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Living Rev. Relativity, 15, (2012), 4http://www.livingreviews.org/lrr-2012-4

(Update of lrr-2005-3)

L I V I N G REVIEWS

in relativity

Gravitational-Wave Data Analysis.

Formalism and Sample Applications: The Gaussian Case

Piotr JaranowskiFaculty of Physics

University of Bia lystokLipowa 41

15–424 Bia lystok, Polandemail: [email protected]

Andrzej KrolakInstitute of Mathematics

Polish Academy of SciencesSniadeckich 8

00–956 Warsaw, Polandemail: [email protected]

Accepted on 14 February 2012Published on 9 March 2012

Abstract

The article reviews the statistical theory of signal detection in application to analysis ofdeterministic gravitational-wave signals in the noise of a detector. Statistical foundations forthe theory of signal detection and parameter estimation are presented. Several tools neededfor both theoretical evaluation of the optimal data analysis methods and for their practicalimplementation are introduced. They include optimal signal-to-noise ratio, Fisher matrix,false alarm and detection probabilities, ℱ-statistic, template placement, and fitting factor.These tools apply to the case of signals buried in a stationary and Gaussian noise. Algorithmsto efficiently implement the optimal data analysis techniques are discussed. Formulas are givenfor a general gravitational-wave signal that includes as special cases most of the deterministicsignals of interest.

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9 March 2012: Material of the previous version of the review was partially reorganized andupdated, 46 new references were added.

1. Section 2 was rewritten and extended, several new references were added.

2. Some parts of the former Section 4 were moved to the present Section 3, which is now abrief general introduction to the statistical theory of signal detection and of estimation of signalsparameters. Some new references were added.

3. The present Section 4 is a partially rewritten (using some new, more convenient notation) andextended version of the former Sections 4.3 – 4.9. The gravitational-wave signal considered herewas generalized from a 4-amplitude-parameter case to an n-amplitude-parameter case, where nis arbitrary. New Section 4.1.1 about targeted searches was added, and new Section 4.4.1 on thecovering problem was created with references to constructions of various grids of templates forsearches of continuous gravitational waves.

4. The present Section 5 is an expanded version of the former Section 4.10 with addition of severalrecent references.

5. The present Section 6 is an expanded version of the former Section 4.11 with new discussion ofoptimal filtering for non-stationary data and description of a test (Grubbs’ test) to detect outliersin data.


Contents

1 Introduction 5

2 Response of a Detector to a Gravitational Wave 72.1 Doppler shift between freely falling particles . . . . . . . . . . . . . . . . . . . . . . 72.2 Long-wavelength approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Solar-system-based detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 General form of the response function . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Statistical Theory of Signal Detection 143.1 Hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.1 Bayesian approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.2 Minimax approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.3 Neyman–Pearson approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.4 Likelihood ratio test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 The matched filter in Gaussian noise . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3.1 Bayesian estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.2 Maximum a posteriori probability estimation . . . . . . . . . . . . . . . . . 173.3.3 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 Fisher information and Cramer–Rao bound . . . . . . . . . . . . . . . . . . . . . . 18

4 Maximum-likelihood Detection in Gaussian Noise 194.1 The ℱ-statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1.1 Targeted searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Signal-to-noise ratio and the Fisher matrix . . . . . . . . . . . . . . . . . . . . . . 204.3 False alarm and detection probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3.1 False alarm and detection probabilities for known intrinsic parameters . . . 224.3.2 False alarm probability for unknown intrinsic parameters . . . . . . . . . . 234.3.3 Detection probability for unknown intrinsic parameters . . . . . . . . . . . 25

4.4 Number of templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4.1 Covering problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.5 Suboptimal filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.6 Algorithms to calculate the ℱ-statistic . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.6.1 The two-step procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.6.2 Evaluation of the ℱ-statistic . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.7 Accuracy of parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.7.1 Fisher-matrix-based assessments . . . . . . . . . . . . . . . . . . . . . . . . 304.7.2 Comparison with the Cramer–Rao bound . . . . . . . . . . . . . . . . . . . 31

4.8 Upper limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Network of Detectors 33

6 Non-stationary, Non-Gaussian, and Non-linear Data 35

7 Acknowledgments 38

References 39

Gravitational-Wave Data Analysis. Formalism and Sample Applications: The Gaussian Case 5

1 Introduction

In this review we consider the problem of detection of deterministic gravitational-wave signalsin the noise of a detector and the question of estimation of their parameters. The examplesof deterministic signals are gravitational waves from rotating neutron stars, coalescing compactbinaries, and supernova explosions. The case of detection of stochastic gravitational-wave signalsin the noise of a detector is reviewed in [8]. A very powerful method to detect a signal in noise thatis optimal by several criteria consists of correlating the data with the template that is matched tothe expected signal. This matched-filtering technique is a special case of the maximum likelihooddetection method. In this review we describe the theoretical foundation of the method and weshow how it can be applied to the case of a very general deterministic gravitational-wave signalburied in a stationary and Gaussian noise.

Early gravitational-wave data analysis was concerned with the detection of bursts originatingfrom supernova explosions [144]. It involved analysis of the coincidences among the detectors [70].With the growing interest in laser interferometric gravitational-wave detectors that are broadbandit was realized that sources other than supernovae can also be detectable [135] and that they canprovide a wealth of astrophysical information [122, 77]. For example, the analytic form of thegravitational-wave signal produced during the inspiral phase of a compact binary coalescence isknown in terms of a few parameters to a good approximation (see, e.g., [30] and Section 2.4 of [66]).Consequently one can detect such a signal by correlating the data with the predicted waveform(often called the template) and maximizing the correlation with respect to the parameters ofthe waveform. Using this method one can pick up a weak signal from the noise by building alarge signal-to-noise ratio over a wide bandwidth of the detector [135]. This observation has ledto rapid development of the theory of gravitational-wave data analysis. It became clear that thedetectability of sources is determined by optimal signal-to-noise ratio, which is the power spectrumof the signal divided by the power spectrum of the noise integrated over the bandwidth of thedetector.

An important landmark was a workshop entitled Gravitational Wave Data Analysis held inDyffryn House and Gardens, St. Nicholas near Cardiff, in July 1987 [123]. The meeting acquaintedphysicists interested in analyzing gravitational-wave data with the basics of the statistical theoryof signal detection and its application to detection of gravitational-wave sources. As a result ofsubsequent studies, the Fisher information matrix was introduced to the theory of the analysis ofgravitational-wave data [50, 76]. The diagonal elements of the Fisher matrix give lower boundson the variances of the estimators of the parameters of the signal and can be used to assess thequality of astrophysical information that can be obtained from detections of gravitational-wavesignals [41, 75, 25]. It was also realized that the application of matched-filtering to some sources,notably to continuous sources originating from neutron stars, will require extraordinary largecomputing resources. This gave a further stimulus to the development of optimal and efficientalgorithms and data analysis methods [124].

A very important development was the work by Cutler et al. [43] where it was realized that forthe case of coalescing binaries matched filtering was sensitive to very small post-Newtonian effectsof the waveform. Thus, these effects can be detected. This leads to a much better verification ofEinstein’s theory of relativity and provides a wealth of astrophysical information that would make alaser interferometric gravitational-wave detector a true astronomical observatory complementary tothose utilizing the electromagnetic spectrum. As further development of the theory, methods wereintroduced to calculate the quality of suboptimal filters [13], to calculate the number of templatesrequired to do a search using matched-filtering [102], to determine the accuracy of templatesrequired [33], and to calculate the false alarm probability and thresholds [68]. An important pointis the reduction of the number of parameters that one needs to search for in order to detect a signal.Namely estimators of a certain type of parameters, called extrinsic parameters, can be found in

Living Reviews in Relativityhttp://www.livingreviews.org/lrr-2012-4


6 Piotr Jaranowski and Andrzej Krolak

a closed analytic form and consequently eliminated from the search. Thus, a computationally-intensive search need only be performed over a reduced set of intrinsic parameters [76, 68, 78].

Techniques reviewed in this paper have been used in the data analysis of prototypes of gravita-tional-wave detectors [100, 99, 12] and in the data analysis of gravitational-wave detectors currentlyin operation [133, 23, 4, 3, 2].




2 Response of a Detector to a Gravitational Wave

There are two main methods of detecting gravitational waves currently in use. One method is tomeasure changes induced by gravitational waves on the distances between freely-moving test massesusing coherent trains of electromagnetic waves. The other method is to measure the deformationof large masses at their resonance frequencies induced by gravitational waves. The first idea isrealized in laser interferometric detectors (both Earth-based [105, 141, 53] and space-borne [85,137] antennas) and Doppler tracking experiments [14], whereas the second idea is implemented inresonant mass detectors (see, e.g., [22]).

2.1 Doppler shift between freely falling particles

We start by describing the change of a photon’s frequency caused by a passing gravitational waveand registered by particles (representing different parts of a gravitational-wave detector) freelyfalling in the field of the gravitational wave. The detailed derivation of the formulae we showhere can be found in Chapter 5 of [66] (see also [47, 15, 120]). An equivalent derivation of theresponse of test masses to gravitational waves in the local Lorentz gauge (without making use ofthe long-wavelength approximation) is given in [113].

We employ here the transverse traceless (TT) coordinate system (more about the TT gauge canbe found, e.g., in Section 35.4 of [91] or in Section 1.3 of [66]). A spacetime metric describing a planegravitational wave traveling in the +𝑧 direction of the TT coordinate system (with coordinates𝑥0 ≡ 𝑐𝑡, 𝑥1 ≡ 𝑥, 𝑥2 ≡ 𝑦, 𝑥3 ≡ 𝑧), is described by the line element

d𝑠2 = −𝑐2d𝑡2 +

(1 + ℎ+

(𝑡− 𝑧

𝑐

))d𝑥2 +

(1− ℎ+

(𝑡− 𝑧

𝑐

))d𝑦2 + 2ℎ×

(𝑡− 𝑧

𝑐

)d𝑥d𝑦 + d𝑧2, (1)

where ℎ+ and ℎ× are the two independent polarizations of the wave. We assume that the wave isweak, i.e., for any instant of time 𝑡,

|ℎ+(𝑡)| ≪ 1, |ℎ×(𝑡)| ≪ 1. (2)

We will neglect all terms of order ℎ2 or higher. The form of the line element (1) implies that thefunctions ℎ+(𝑡) and ℎ×(𝑡) describe the wave-induced perturbation of the flat Minkowski metric atthe origin of the TT coordinate system (where 𝑥 = 𝑦 = 𝑧 = 0). It is convenient to introduce thethree-dimensional matrix of the spatial metric perturbation produced by the gravitational wave(at the coordinate system’s origin),

H(𝑡) :=

⎛⎝ℎ+(𝑡) ℎ×(𝑡) 0ℎ×(𝑡) −ℎ+(𝑡) 00 0 0

⎞⎠ . (3)

Let two particles freely fall in the field (1) of the gravitational wave, and let their spatialcoordinates remain constant, so the particles’ world lines are described by equations

𝑡(𝜏𝑎) = 𝜏𝑎, 𝑥(𝜏𝑎) = 𝑥𝑎, 𝑦(𝜏𝑎) = 𝑦𝑎, 𝑧(𝜏𝑎) = 𝑧𝑎, 𝑎 = 1, 2, (4)

where (𝑥𝑎, 𝑦𝑎, 𝑧𝑎) are spatial coordinates of the 𝑎th particle and 𝜏𝑎 is its proper time. These twoparticles measure, in their proper reference frames, the frequency of the same photon travelingalong a null geodesic 𝑥𝛼 = 𝑥𝛼(𝜆), where 𝜆 is some affine parameter. The coordinate time, at whichthe photon’s frequency is measured by the 𝑎th particle, is equal to 𝑡𝑎 (𝑎 = 1, 2); we assume that𝑡2 > 𝑡1. Let us introduce the coordinate time duration 𝑡12 of the photon’s trip and the Euclideancoordinate distance 𝐿12 between the particles:

𝑡12 := 𝑡2 − 𝑡1, 𝐿12 :=√

(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 + (𝑧2 − 𝑧1)2. (5)




Let us also introduce the 3-vector n of unit Euclidean length directed along the line connecting thetwo particles. We arrange the components of this vector into the column 3× 1 matrix n (thus, wedistinguish here the 3-vector n from its components being the elements of the matrix n; the same3-vector can be decomposed into components in different spatial coordinate systems):

n := (cos𝛼, cos𝛽, cos 𝛾)T =

⎛⎝cos𝛼cos𝛽cos 𝛾

⎞⎠ , (6)

where the superscript T denotes matrix transposition. If one neglects the spacetime curvaturecaused by the gravitational wave, then 𝛼, 𝛽, and 𝛾 are the angles between the path of the photonin the 3-space and the coordinate axis 𝑥, 𝑦, or 𝑧, respectively (obviously, 𝛼, 𝛽, 𝛾 ∈ ⟨0;𝜋⟩ andcos2 𝛼 + cos2 𝛽 + cos2 𝛾 = 1). Let us denote the value of the frequency registered by the 𝑎thparticle by 𝜈𝑎 (𝑎 = 1, 2) and let us finally define the relative change of the photon’s frequencies,

𝑦12 :=𝜈2𝜈1

− 1. (7)

Then, it can be shown (see Chapter 5 of [66] for details) that the frequency ratio 𝑦12 can be written[making use of the quantities introduced in Eqs. (3) and (5) – (6)] as follows (the dot means herematrix multiplication):

𝑦12 =1

2(1− cos 𝛾)nT ·

(H(𝑡1 −

𝑧1𝑐

)− H

(𝑡1 −

𝑧1𝑐

+ (1− cos 𝛾)𝐿12

𝑐

))· n+𝒪

(ℎ2). (8)

It is convenient to introduce the unit 3-vector k directed from the origin of the coordinatesystem to the source of the gravitational wave. In the coordinate system adopted by us the waveis traveling in the +𝑧 direction. Therefore, the components of the 3-vector k, arranged into thecolumn matrix k, are

k = (0, 0,−1)T. (9)

The positions of the particles with respect to the origin of the coordinate system we describe bythe 3-vectors x𝑎 (𝑎 = 1, 2), the components of which we put into the column matrices x𝑎:

x𝑎 = (𝑥𝑎, 𝑦𝑎, 𝑧𝑎)T, 𝑎 = 1, 2. (10)

Making use of Eqs. (9) – (10) we rewrite the basic formula (8) in the following form

𝑦12 =

nT ·(H(𝑡1 +

kT · x1𝑐

)− H

(𝑡1 +

𝐿12

𝑐+

kT · x2𝑐

))· n

2(1 + kT · n)+𝒪

(ℎ2). (11)

To obtain the response for all currently working and planned detectors it is enough to considera configuration of three particles shown in Figure 1. Two particles model a Doppler trackingexperiment, where one particle is the Earth and the other is a distant spacecraft. Three particlesmodel a ground-based laser interferometer, where the masses are suspended from seismically-isolated supports or a space-borne interferometer, where the three test masses are shielded insatellites driven by drag-free control systems. In Figure 1 we have introduced the following notation:O denotes the origin of the TT coordinate system related to the passing gravitational wave, x𝑎

(𝑎 = 1, 2, 3) are 3-vectors joining O and the particles, n𝑎 and 𝐿𝑎 (𝑎 = 1, 2, 3) are, respectively,3-vectors of unit Euclidean length along the lines joining the particles and the coordinate Euclideandistances between the particles, where 𝑎 is the label of the opposite particle. We still assume thatthe spatial coordinates of the particles do not change in time.




1

2

3

O

n1

n2

n3 L1

L2

L3

x1

x2

x3

Figure 1: Schematic configuration of three freely-falling particles as a detector of gravitational waves.The particles are labelled 1, 2, and 3, their positions with respect to the origin O of the coordinate systemare given by 3-vectors x𝑎 (𝑎 = 1, 2, 3). The Euclidean separations between the particles are denoted by𝐿𝑎, where the index 𝑎 corresponds to the opposite particle. The unit 3-vectors n𝑎 point between pairs ofparticles, with the orientation indicated.

Let us denote by 𝜈0 the frequency of the coherent beam used in the detector (laser light in thecase of an interferometer and radio waves in the case of Doppler tracking). Let the particle 1 emitthe photon with frequency 𝜈0 at the moment 𝑡0 towards the particle 2, which registers the photonwith frequency 𝜈′ at the moment 𝑡′ = 𝑡0 + 𝐿3/𝑐+𝒪(ℎ). The photon is immediately transponded(without change of frequency) back to the particle 1, which registers the photon with frequency 𝜈at the moment 𝑡 = 𝑡0 + 2𝐿3/𝑐+𝒪(ℎ). We express the relative changes of the photon’s frequency𝑦12 := (𝜈′ − 𝜈0)/𝜈0 and 𝑦21 := (𝜈 − 𝜈′)/𝜈′ as functions of the instant of time 𝑡. Making use ofEq. (11) we obtain

𝑦12(𝑡) =1

2(1− kT · n3)nT3 ·

(H(𝑡− 2𝐿3

𝑐+

kT · x1𝑐

)− H

(𝑡− 𝐿3

𝑐+

kT · x2𝑐

))· n3 +𝒪

(ℎ2),(12a)

𝑦21(𝑡) =1

2(1 + kT · n3)nT3 ·

(H(𝑡− 𝐿3

𝑐+

kT · x2𝑐

)− H

(𝑡+

kT · x1𝑐

))· n3 +𝒪

(ℎ2). (12b)

The total frequency shift 𝑦121 := (𝜈 − 𝜈0)/𝜈0 of the photon during its round trip can be computedfrom the one-way frequency shifts 𝑦12 and 𝑦21 given above:

𝑦121 =𝜈

𝜈0− 1 =

𝜈

𝜈′𝜈′

𝜈0− 1 = (𝑦21 + 1)(𝑦12 + 1)− 1 = 𝑦12 + 𝑦21 +𝒪

(ℎ2). (13)

2.2 Long-wavelength approximation

Let 𝐿 be the size of the detector and 𝜆 := 𝜆/(2𝜋) be the reduced wavelength of the gravitationalwave impinging on the detector. In the long-wavelength approximation the condition 𝜆 ≫ 𝐿 isfulfilled. The angular frequency of the wave equals 𝜔 = 𝑐/𝜆. Time delays caused by the finitespeed of the wave propagating across the detector are of order Δ𝑡 ∼ 𝐿/𝑐, but

𝜔Δ𝑡 ∼ 𝐿

𝜆≪ 1, (14)

so time delays across the detector are much shorter than the period of the gravitational wave andcan be neglected. It means that with a good accuracy the gravitational-wave field can be treatedas being uniform (but time-dependent) in the space region that covers the entire detector. To




detect gravitational waves with some dominant angular frequency 𝜔 one must collect data overtime intervals longer (sometimes much longer) than the gravitational-wave period. This impliesthat in Eq. (8) for the relative frequency shift, the typical value of the quantity 𝑡 := 𝑡1 − 𝑧1/𝑐 willbe much larger than the retardation time Δ𝑡 := 𝐿12/𝑐. Therefore, we can expand this equationwith respect to Δ𝑡 and keep terms only linear in Δ𝑡. After doing this one obtains (see Section 5.3in [66] for more details):

𝑦12 = −𝐿12

2𝑐nT · H

(𝑡1 −

𝑧1𝑐

)· n+𝒪

(ℎ2,Δ𝑡2

), (15)

where overdot denotes differentiation with respect to time.For the configuration of particles shown in Figure 1, the relative frequency shifts 𝑦12 and 𝑦21

given by Eqs. (12) can be written, by virtue of the formula (15), in the form

𝑦12(𝑡) = 𝑦21(𝑡) = −𝐿3

2𝑐nT3 · H

(𝑡+

kT · x1𝑐

)· n3 +𝒪

(ℎ2,Δ𝑡2

), (16)

so that they are equal to each other up to terms 𝒪(ℎ2,Δ𝑡2

). The photon’s total round-trip

frequency shift 𝑦121 [cf. Eq. (13)] is thus equal to

𝑦121(𝑡) = −𝐿3

𝑐nT3 · H

(𝑡+

kT · x1𝑐

)· n3 +𝒪

(ℎ2,Δ𝑡2

). (17)

There are important cases where the long-wavelength approximation is not valid. These includesatellite Doppler tracking measurements and the space-borne LISA detector for gravitational-wavefrequencies larger than a few mHz.

2.3 Solar-system-based detectors

Real gravitational-wave detectors do not stay at rest with respect to the TT coordinate systemrelated to the passing gravitational wave, because they also move in the gravitational field ofthe solar system bodies, as in the case of the LISA spacecraft, or are fixed to the surface of theEarth, as in the case of Earth-based laser interferometers or resonant bar detectors. Let us choosethe origin O of the TT coordinate system employed in Section 2.1 to coincide with the solarsystem barycenter (SSB). The motion of the detector with respect to the SSB will modulate thegravitational-wave signal registered by the detector. One can show that as far as the velocities ofthe particles (modeling the detector’s parts) with respect to the SSB are non-relativistic, which isthe case for all existing or planned detectors, Eqs. (12) can still be used, provided the 3-vectors x𝑎

and n𝑎 (𝑎 = 1, 2, 3) will be interpreted as made of the time-dependent components describing themotion of the particles with respect to the SSB.

It is often convenient to introduce the proper reference frame of the detector with coordinates(��𝛼). Because the motion of this frame with respect to the SSB is non-relativistic, we can as-sume that the transformation between the SSB-related coordinates (𝑥𝛼) and the detector’s properreference frame coordinates (��𝛼) has the form

𝑡 = 𝑡, ��𝑖(𝑡, 𝑥𝑘) = ��𝑖𝑂(𝑡) + O𝑖𝑗(𝑡)𝑥

𝑗 , (18)

where the functions ��𝑖𝑂(𝑡) describe the motion of the origin 𝑂 of the proper reference frame with

respect to the SSB, and the functions O𝑖𝑗(𝑡) account for the different orientations of the spatial

axes of the two reference frames. One can compute some of the quantities entering Eqs. (12) in

the detector’s coordinates rather than in the TT coordinates. For instance, the matrix H of thewave-induced spatial metric perturbation in the detector’s coordinates is related to the matrix H




of the spatial metric perturbation produced by the wave in the TT coordinate system throughequation H(𝑡) = (O(𝑡)−1)T · H(𝑡) · O(𝑡)−1, (19)

where the matrix O has elements O𝑖𝑗 . If the transformation matrix O is orthogonal, then O−1 = OT,

and Eq. (19) simplifies to H(𝑡) = O(𝑡) · H(𝑡) · O(𝑡)T. (20)

See [31, 54, 68, 78] for more details.For a ground-based laser-interferometric detector, the long-wavelength approximation can be

employed (however, see [27, 115, 114] for a discussion of importance of high-frequency corrections,which modify the interferometer response function computed within the long-wavelength approx-imation). In the case of an interferometer in a standard Michelson and equal-arm configuration(such configurations can be represented by Figure 1 with the particle 1 corresponding to the cornerstation of the interferometer and with 𝐿2 = 𝐿3 = 𝐿), the observed relative frequency shift Δ𝜈(𝑡)/𝜈0is equal to the difference of the round-trip frequency shifts in the two detector’s arms [136]:

Δ𝜈(𝑡)

𝜈0= 𝑦131(𝑡)− 𝑦121(𝑡). (21)

Let (𝑥d, 𝑦d, 𝑧d) be the components (with respect to the TT coordinate system) of the 3-vector rdconnecting the origin of the TT coordinate system with the corner station of the interferometer.Then x1 = (𝑥d, 𝑦d,𝑧d)

T, kT ·x1 = −𝑧d, and, making use of Eq. (17), the relative frequency shift (21)can be written as

Δ𝜈(𝑡)

𝜈0=

𝐿

𝑐

(nT2 · H

(𝑡− 𝑧d

𝑐

)· n2 − nT3 · H

(𝑡− 𝑧d

𝑐

)· n3). (22)

The difference Δ𝜑(𝑡) of the phase fluctuations measured, say, by a photo detector, is related to thecorresponding relative frequency fluctuations Δ𝜈(𝑡) by

Δ𝜈(𝑡)

𝜈0=

1

2𝜋𝜈0

dΔ𝜑(𝑡)

d𝑡. (23)

One can integrate Eq. (22) to write the phase change Δ𝜑(𝑡) as

Δ𝜑(𝑡) = 4𝜋𝜈0𝐿ℎ(𝑡), (24)

where the dimensionless function ℎ,

ℎ(𝑡) :=1

2

(nT2 · H

(𝑡− 𝑧d

𝑐

)· n2 − nT3 · H

(𝑡− 𝑧d

𝑐

)· n3), (25)

is the response function of the interferometric detector to a plane gravitational wave in the long-wavelength approximation. To get Eqs. (24) – (25) directly from Eqs. (22) – (23) one should assumethat the quantities n2, n3, and 𝑧d [entering Eq. (22)] do not depend on time 𝑡. But the formu-lae (24) – (25) can also be used in the case when those quantities are time dependent, providedthe velocities x𝑎 of the detector’s parts with respect to the SSB are non-relativistic. The error wemake in such cases is on the order of 𝒪(ℎ𝑣), where 𝑣 is a typical value of the velocities x𝑎. Thus,the response function of the Earth-based interferometric detector equals

ℎ(𝑡) =1

2

(n2(𝑡)

T · H(𝑡− 𝑧d(𝑡)

𝑐

)· n2(𝑡)− n3(𝑡)

T · H(𝑡− 𝑧d(𝑡)

𝑐

)· n3(𝑡)

), (26)




where all quantities here are computed in the SSB-related TT coordinate system. The sameresponse function can be written in terms of a detector’s proper-reference-frame quantities asfollows

ℎ(𝑡) =1

2

(nT2 · H

(𝑡− 𝑧d(𝑡)

𝑐

)· n2 − nT3 · H


𝑐

)· n3), (27)

where the matrices H and H are related to each other by means of formula (20). In Eq. (27) theproper-reference-frame components n2 and n3 of the unit vectors directed along the interferometerarms can be treated as constant (i.e., time independent) quantities.

From Eqs. (26) and (3) it follows that the response function ℎ is a linear combination of thetwo wave polarizations ℎ+ and ℎ×, so it can be written as

ℎ(𝑡) = 𝐹+(𝑡)ℎ+


𝑐

)+ 𝐹×(𝑡)ℎ×


𝑐

). (28)

The functions 𝐹+ and 𝐹× are the interferometric beam-pattern functions. They depend on thelocation of the detector on Earth, the position of the gravitational-wave source in the sky, and thepolarization angle of the wave (this angle describes the orientation, with respect to the detector,of the axes relative to which the plus and cross polarizations of the wave are defined, see, e.g.,Figure 9.2 in [135]). Derivation of the explicit formulae for the interferometric beam patterns 𝐹+

and 𝐹× can be found, e.g., in Appendix C of [66].

In the long-wavelength approximation, the response function of the interferometric detectorcan be derived directly from the equation of geodesic deviation [126]. Then the response is definedas the relative difference between the wave-induced changes of the proper lengths of the two arms,i.e., ℎ(𝑡) := (Δ��2(𝑡) − Δ��3(𝑡))/��0, where ��0 + Δ��2(𝑡) and ��0 + Δ��3(𝑡) are the instantaneousvalues of the proper lengths of the interferometer’s arms and ��0 is the unperturbed proper lengthof these arms.

In the case of an Earth-based resonant-bar detector the long-wavelength approximation is veryaccurate and the dimensionless detector’s response function can be defined as ℎ(𝑡) := Δ��(𝑡)/��0,where Δ��(𝑡) is the wave-induced change of the proper length ��0 of the bar. The response functionℎ can be computed in terms of the detector’s proper-reference-frame quantities from the formula(see, e.g., Section 9.5.2 in [135])

ℎ(𝑡) = nT · H(𝑡− 𝑧d(𝑡)

𝑐

)· n, (29)

where the column matrix n consists of the components (computed in the proper reference frameof the detector) of the unit vector n directed along the symmetry axis of the bar. The responsefunction (29) can be written as a linear combination of the wave polarizations ℎ+ and ℎ×, i.e.,the formula (28) is also valid for the resonant-bar response function but with some bar-patternfunctions 𝐹+ and 𝐹× different from the interferometric beam-pattern functions. Derivation of theexplicit form of the bar patterns can be found, e.g., in Appendix C of [66].

2.4 General form of the response function

In many cases of interest the response function of the detector to a plane gravitational wave canbe written as a linear combination of 𝑛 waveforms ℎ𝑘(𝑡; 𝜉) [which all depend on the same set ofparameters 𝜉 = (𝜉1, . . . , 𝜉𝑚)] with constant amplitudes 𝑎𝑘 (𝑘 = 1, . . . , 𝑛),

ℎ(𝑡;𝜃) =𝑛∑

𝑘=1

𝑎𝑘 ℎ𝑘(𝑡; 𝜉), (30)




where the vector 𝜃 collects all the signal’s parameters. It is convenient to introduce column matrices

a :=

⎛⎜⎝𝑎1...𝑎𝑛

⎞⎟⎠ , h(𝑡; 𝜉) :=

⎛⎜⎝ℎ1(𝑡; 𝜉)...

ℎ𝑛(𝑡; 𝜉)

⎞⎟⎠ . (31)

Then one can briefly write 𝜃 = (a, 𝜉) and the response (30) can be written as

ℎ(𝑡;𝜃) = aT · h(𝑡; 𝜉). (32)

The functions ℎ𝑘 (𝑘 = 1, . . . , 𝑛) are independent of the parameters a. The parameters a are calledextrinsic (or amplitude) parameters whereas the parameters 𝜉 are called intrinsic.

Eq. (32) with 𝑛 = 4 is a model of the response of the space-based detector LISA to gravitationalwaves from a binary system [78]. Also for 𝑛 = 4 the same equation models the response of a ground-based detector to a continuous source of gravitational waves like a rotating neutron star [68].For ground-based detectors the long-wavelength approximation can be applied and within thisapproximation the functions ℎ𝑘 (𝑘 = 1, . . . , 4) are given by

ℎ1(𝑡; 𝜉) = 𝑢(𝑡; 𝜉) cos𝜑(𝑡; 𝜉),

ℎ2(𝑡; 𝜉) = 𝑣(𝑡; 𝜉) cos𝜑(𝑡; 𝜉),

ℎ3(𝑡; 𝜉) = 𝑢(𝑡; 𝜉) sin𝜑(𝑡; 𝜉),

ℎ4(𝑡; 𝜉) = 𝑣(𝑡; 𝜉) sin𝜑(𝑡; 𝜉),

(33)

where 𝜑(𝑡; 𝜉) is the phase modulation of the signal and 𝑢(𝑡; 𝜉) and 𝑣(𝑡; 𝜉) are slowly varyingamplitude modulations. The gravitational-wave signal from spinning neutron stars may consistof several components of the form (32). For short observation times over which the amplitudemodulation functions are nearly constant, the response of the ground-based detector can furtherbe approximated by

ℎ(𝑡;𝐴0, 𝜑0, 𝜉) = 𝐴0 𝑔(𝑡; 𝜉) cos (𝜑(𝑡; 𝜉)− 𝜑0) , (34)

where 𝐴0 and 𝜑0 are constant amplitude and initial phase, respectively, and 𝑔(𝑡; 𝜉) is a slowlyvarying function of time. Eq. (34) is a good model for the response of a detector to the gravitationalwave from a coalescing compact binary system [135, 30]. We would like to stress that not alldeterministic gravitational-wave signals may be cast into the general form (32).




3 Statistical Theory of Signal Detection

The gravitational-wave signal will be buried in the noise of the detector and the data from thedetector will be a random (or stochastic) process. Consequently, the problem of extracting thesignal from the noise is a statistical one. The basic idea behind signal detection is that the presenceof the signal changes the statistical characteristics of the data 𝑥, in particular its probabilitydistribution. When the signal is absent the data have probability density function (pdf) 𝑝0(𝑥),and when the signal is present the pdf is 𝑝1(𝑥).

A thorough introduction to probability theory and mathematical statistics can be found, e.g.,in [51, 130, 131, 101]. A full exposition of statistical theory of signal detection that is only outlinedhere can be found in the monographs [147, 74, 143, 139, 87, 58, 107]. A general introductionto stochastic processes is given in [145] and advanced treatment of the subject can be foundin [84, 146]. A concise introduction to the statistical theory of signal detection and time seriesanalysis is contained in Chapters 3 and 4 of [66].

3.1 Hypothesis testing

The problem of detecting the signal in noise can be posed as a statistical hypothesis testing problem.The null hypothesis 𝐻0 is that the signal is absent from the data and the alternative hypothesis 𝐻1

is that the signal is present. A hypothesis test (or decision rule) 𝛿 is a partition of the observationset into two subsets, ℛ and its complement ℛ′. If data are in ℛ we accept the null hypothesis,otherwise we reject it. There are two kinds of errors that we can make. A type I error is choosinghypothesis 𝐻1 when 𝐻0 is true and a type II error is choosing 𝐻0 when 𝐻1 is true. In signaldetection theory the probability of a type I error is called the false alarm probability, whereas theprobability of a type II error is called the false dismissal probability. 1−(false dismissal probability)is the probability of detection of the signal. In hypothesis testing theory, the probability of a typeI error is called the significance of the test, whereas 1− (probability of type II error) is called thepower of the test.

The problem is to find a test that is in some way optimal. There are several approaches tofinding such a test. The subject is covered in detail in many books on statistics, for example,see [72, 51, 80, 83].

3.1.1 Bayesian approach

In the Bayesian approach we assign costs to our decisions; in particular we introduce positivenumbers 𝐶𝑖𝑗 , 𝑖, 𝑗 = 0, 1, where 𝐶𝑖𝑗 is the cost incurred by choosing hypothesis 𝐻𝑖 when hypothesis𝐻𝑗 is true. We define the conditional risk 𝑅 of a decision rule 𝛿 for each hypothesis as

𝑅𝑗(𝛿) = 𝐶0𝑗𝑃𝑗(ℛ) + 𝐶1𝑗𝑃𝑗(ℛ′), 𝑗 = 0, 1, (35)

where 𝑃𝑗 is the probability distribution of the data when hypothesis 𝐻𝑗 is true. Next, we assignprobabilities 𝜋0 and 𝜋1 = 1− 𝜋0 to the occurrences of hypotheses 𝐻0 and 𝐻1, respectively. Theseprobabilities are called a priori probabilities or priors. We define the Bayes risk as the overallaverage cost incurred by the decision rule 𝛿:

𝑟(𝛿) = 𝜋0𝑅0(𝛿) + 𝜋1𝑅1(𝛿). (36)

Finally we define the Bayes rule as the rule that minimizes the Bayes risk 𝑟(𝛿).

3.1.2 Minimax approach

Very often in practice we do not have control over or access to the mechanism generating the stateof nature and we are not able to assign priors to various hypotheses. In such a case one criterion




is to seek a decision rule that minimizes, over all 𝛿, the maximum of the conditional risks, 𝑅0(𝛿)and 𝑅1(𝛿). A decision rule that fulfills that criterion is called a minimax rule.

3.1.3 Neyman–Pearson approach

In many problems of practical interest the imposition of a specific cost structure on the decisionsmade is not possible or desirable. The Neyman–Pearson approach involves a trade-off between thetwo types of errors that one can make in choosing a particular hypothesis. The Neyman–Pearsondesign criterion is to maximize the power of the test (probability of detection) subject to a chosensignificance of the test (false alarm probability).

3.1.4 Likelihood ratio test

It is remarkable that all three very different approaches – Bayesian, minimax, and Neyman–Pearson– lead to the same test called the likelihood ratio test [44]. The likelihood ratio Λ is the ratio ofthe pdf when the signal is present to the pdf when it is absent:

Λ(𝑥) :=𝑝1(𝑥)

𝑝0(𝑥). (37)

We accept the hypothesis 𝐻1 if Λ > 𝑘, where 𝑘 is the threshold that is calculated from the costs𝐶𝑖𝑗 , priors 𝜋𝑖, or the significance of the test depending on which approach is being used.

3.2 The matched filter in Gaussian noise

Let ℎ be the gravitational-wave signal we are looking for and let 𝑛 be the detector’s noise. Forconvenience we assume that the signal ℎ is a continuous function of time 𝑡 and that the noise 𝑛 isa continuous random process. Results for the discrete-in-time data that we have in practice canthen be obtained by a suitable sampling of the continuous-in-time expressions. Assuming that thenoise is additive the data 𝑥 can be written as

𝑥(𝑡) = 𝑛(𝑡) + ℎ(𝑡). (38)

The autocorrelation function of the noise 𝑛 is defined as

𝐾𝑛(𝑡, 𝑡′) := E [𝑛(𝑡)𝑛(𝑡′)] , (39)

where E denotes the expectation value.Let us further assume that the detector’s noise 𝑛 is a zero-mean and Gaussian random pro-

cess. It can then be shown that the logarithm of the likelihood function is given by the followingCameron–Martin formula

log Λ[𝑥] =

∫ 𝑇o

0

𝑞(𝑡)𝑥(𝑡) d𝑡− 1

2

∫ 𝑇o

0

𝑞(𝑡)ℎ(𝑡) d𝑡, (40)

where ⟨0;𝑇o⟩ is the time interval over which the data was collected and the function 𝑞 is the solutionof the integral equation

ℎ(𝑡) =

∫ 𝑇o

0

𝐾𝑛(𝑡, 𝑡′)𝑞(𝑡′) d𝑡. (41)

For stationary noise, its autocorrelation function (39) depends on times 𝑡 and 𝑡′ only throughthe difference 𝑡− 𝑡′. It implies that there exists then an even function 𝜅𝑛 of one variable such that

E[𝑛(𝑡)𝑛(𝑡′)] = 𝜅𝑛(𝑡− 𝑡′). (42)




Spectral properties of stationary noise are described by its one-sided spectral density, defined fornon-negative frequencies 𝑓 ≥ 0 through relation

𝑆𝑛(𝑓) = 2

∫ ∞

−∞𝜅𝑛(𝑡)𝑒

2𝜋𝑖𝑓𝑡d𝑡. (43)

For negative frequencies 𝑓 < 0, by definition, 𝑆𝑛(𝑓) = 0. The spectral density 𝑆𝑛 can also bedetermined by correlations between the Fourier components of the detector’s noise

E [��(𝑓)��*(𝑓 ′)] =1

2𝑆𝑛(|𝑓 |)𝛿(𝑓 − 𝑓 ′), −∞ < 𝑓, 𝑓 ′ < ∞. (44)

For the case of stationary noise with one-sided spectral density 𝑆𝑛, it is convenient to definethe scalar product (𝑥|𝑦) between any two waveforms 𝑥 and 𝑦,

(𝑥|𝑦) := 4ℜ∫ ∞

0

��(𝑓)𝑦*(𝑓)

𝑆𝑛(𝑓)d𝑓, (45)

where ℜ denotes the real part of a complex expression, the tilde denotes the Fourier transform,and the asterisk is complex conjugation. Making use of this scalar product, the log likelihoodfunction (40) can be written as

log Λ[𝑥] = (𝑥|ℎ)− 1

2(ℎ|ℎ). (46)

From the expression (46) we see immediately that the likelihood ratio test consists of correlating thedata 𝑥 with the signal ℎ that is present in the noise and comparing the correlation to a threshold.Such a correlation is called the matched filter. The matched filter is a linear operation on the data.

An important quantity is the optimal signal-to-noise ratio 𝜌 defined by

𝜌 :=√

(ℎ|ℎ). (47)

By means of Eq. (45) it can be written as

𝜌2 = 4

∫ ∞

0

|ℎ(𝑓)|2

𝑆𝑛(𝑓)d𝑓. (48)

We see in the following that 𝜌 determines the probability of detection of the signal. The higherthe signal-to-noise ratio the higher the probability of detection.

An interesting property of the matched filter is that it maximizes the signal-to-noise ratio overall linear filters [44]. This property is independent of the probability distribution of the noise.

3.3 Parameter estimation

Very often we know the waveform of the signal that we are searching for in the data in terms ofa finite number of unknown parameters. We would like to find optimal procedures of estimatingthese parameters. An estimator of a parameter 𝜃 is a function 𝜃(𝑥) that assigns to each data set

the “best” guess of the true value of 𝜃. Note that because 𝜃(𝑥) depends on the random data it is arandom variable. Ideally we would like our estimator to be (i) unbiased, i.e., its expectation valueto be equal to the true value of the parameter, and (ii) of minimum variance. Such estimators arerare and in general difficult to find. As in the signal detection there are several approaches to theparameter estimation problem. The subject is exposed in detail in [81, 82]. See also [149] for aconcise account.




3.3.1 Bayesian estimation

We assign a cost function 𝐶(𝜃′, 𝜃) of estimating the true value of 𝜃 as 𝜃′. We then associate with

an estimator 𝜃 a conditional risk or cost averaged over all realizations of data 𝑥 for each value ofthe parameter 𝜃:

𝑅𝜃(𝜃) = E𝜃[𝐶(𝜃, 𝜃)] =

∫𝑋

𝐶(𝜃(𝑥), 𝜃

)𝑝(𝑥, 𝜃) d𝑥, (49)

where 𝑋 is the set of observations and 𝑝(𝑥, 𝜃) is the joint probability distribution of data 𝑥 andparameter 𝜃. We further assume that there is a certain a priori probability distribution 𝜋(𝜃) ofthe parameter 𝜃. We then define the Bayes estimator as the estimator that minimizes the averagerisk defined as

𝑟(𝜃) = E[𝑅𝜃(𝜃)] =

∫𝑋

∫Θ

𝐶(𝜃(𝑥), 𝜃

)𝑝(𝑥, 𝜃)𝜋(𝜃) d𝜃 d𝑥, (50)

where E is the expectation value with respect to an a priori distribution 𝜋, and Θ is the set ofobservations of the parameter 𝜃. It is not difficult to show that for a commonly used cost function

𝐶(𝜃′, 𝜃) = (𝜃′ − 𝜃)2, (51)

the Bayesian estimator is the conditional mean of the parameter 𝜃 given data 𝑥, i.e.,

𝜃(𝑥) = E[𝜃|𝑥] =∫Θ

𝜃 𝑝(𝜃|𝑥) d𝜃, (52)

where 𝑝(𝜃|𝑥) is the conditional probability density of parameter 𝜃 given the data 𝑥.

3.3.2 Maximum a posteriori probability estimation

Suppose that in a given estimation problem we are not able to assign a particular cost function𝐶(𝜃′, 𝜃). Then a natural choice is a uniform cost function equal to 0 over a certain interval 𝐼𝜃 ofthe parameter 𝜃. From Bayes theorem [28] we have

𝑝(𝜃|𝑥) = 𝑝(𝑥, 𝜃)𝜋(𝜃)

𝑝(𝑥), (53)

where 𝑝(𝑥) is the probability distribution of data 𝑥. Then, from Eq. (50) one can deduce that foreach data point 𝑥 the Bayes estimate is any value of 𝜃 that maximizes the conditional probability𝑝(𝜃|𝑥). The density 𝑝(𝜃|𝑥) is also called the a posteriori probability density of parameter 𝜃 andthe estimator that maximizes 𝑝(𝜃|𝑥) is called the maximum a posteriori (MAP) estimator. It is

denoted by 𝜃MAP. We find that the MAP estimators are solutions of the following equation

𝜕 log 𝑝(𝑥, 𝜃)

𝜕𝜃= −𝜕 log 𝜋(𝜃)

𝜕𝜃, (54)

which is called the MAP equation.

3.3.3 Maximum likelihood estimation

Often we do not know the a priori probability density of a given parameter and we simply assign toit a uniform probability. In such a case maximization of the a posteriori probability is equivalentto maximization of the probability density 𝑝(𝑥, 𝜃) treated as a function of 𝜃. We call the function𝑙(𝜃, 𝑥) := 𝑝(𝑥, 𝜃) the likelihood function and the value of the parameter 𝜃 that maximizes 𝑙(𝜃, 𝑥) themaximum likelihood (ML) estimator. Instead of the function 𝑙 we can use the function Λ(𝜃, 𝑥) =𝑙(𝜃, 𝑥)/𝑝(𝑥) (assuming that 𝑝(𝑥) > 0). Λ is then equivalent to the likelihood ratio [see Eq. (37)]




when the parameters of the signal are known. Then the ML estimators are obtained by solvingthe equation

𝜕 log Λ(𝜃, 𝑥)

𝜕𝜃= 0, (55)

which is called the ML equation.

3.4 Fisher information and Cramer–Rao bound

It is important to know how good our estimators are. We would like our estimator to have as smalla variance as possible. There is a useful lower bound on variances of the parameter estimatorscalled the Cramer–Rao bound, which is expressed in terms of the Fisher information matrix Γ(𝜃).For the signal ℎ(𝑡;𝜃), which depends on 𝐾 parameters collected into the vector 𝜃 = (𝜃1, . . . , 𝜃𝐾),the components of the matrix Γ(𝜃) are defined as

Γ(𝜃)𝑖𝑗 := E

[𝜕 log Λ[𝑥;𝜃]

𝜕𝜃𝑖

𝜕 log Λ[𝑥;𝜃]

𝜕𝜃𝑗

]= −E

[𝜕2 log Λ[𝑥;𝜃]

𝜕𝜃𝑖 𝜕𝜃𝑗

], 𝑖, 𝑗 = 1, . . . ,𝐾. (56)

The Cramer–Rao bound states that for unbiased estimators the covariance matrix C(𝜃) of theestimators 𝜃 fulfills the inequality

C(𝜃) ≥ Γ(𝜃)−1. (57)

(The inequality A ≥ B for matrices means that the matrix A− B is nonnegative definite.)A very important property of the ML estimators is that asymptotically (i.e., for a signal-to-

noise ratio tending to infinity) they are (i) unbiased, and (ii) they have a Gaussian distributionwith covariance matrix equal to the inverse of the Fisher information matrix.

In the case of Gaussian noise the components of the Fisher matrix are given by

Γ(𝜃)𝑖𝑗 =

(𝜕ℎ(𝑡;𝜃)

𝜕𝜃𝑖

𝜕ℎ(𝑡;𝜃)

𝜕𝜃𝑗

), 𝑖, 𝑗 = 1, . . . ,𝐾, (58)

where the scalar product (·|·) is defined in Eq. (45).




4 Maximum-likelihood Detection in Gaussian Noise

In this section, we study the detection of a deterministic gravitational-wave signal ℎ(𝑡;𝜃) of thegeneral form given by Eq. (32) and the estimation of its parameters 𝜃 using the maximum-likelihood(ML) principle. We assume that the noise 𝑛(𝑡) in the detector is a zero-mean, Gaussian, andstationary random process. The data 𝑥 in the detector, in the case when the gravitational-wavesignal ℎ(𝑡;𝜃) is present, is 𝑥(𝑡;𝜃) = 𝑛(𝑡) + ℎ(𝑡;𝜃). The parameters 𝜃 = (a, 𝜉) of the signal (32)split into extrinsic (or amplitude) parameters a and intrinsic ones 𝜉.

4.1 The ℱ-statistic

For the gravitational-wave signal ℎ(𝑡; a, 𝜉) of the form given in Eq. (32) the log likelihood func-tion (46) can be written as

log Λ[𝑥; a, 𝜉] = aT · N[𝑥; 𝜉]− 1

2aT ·M(𝜉) · a, (59)

where the components of the column 𝑛× 1 matrix N and the square 𝑛× 𝑛 matrix M are given by

𝑁𝑘[𝑥; 𝜉] := (𝑥|ℎ𝑘(𝑡; 𝜉)), 𝑀𝑘𝑙(𝜉) := (ℎ𝑘(𝑡; 𝜉)|ℎ𝑙(𝑡; 𝜉)), 𝑘, 𝑙 = 1, . . . , 𝑛. (60)

The ML equations for the extrinsic parameters a, 𝜕 log Λ[𝑥; a, 𝜉]/𝜕a = 0, can be solved explicitlyto show that the ML estimators a of the parameters a are given by

a[𝑥; 𝜉] = M(𝜉)−1 · N[𝑥; 𝜉]. (61)

Replacing the extrinsic parameters a in Eq. (59) by their ML estimators a, we obtain the reducedlog likelihood function,

ℱ [𝑥; 𝜉] := log Λ[𝑥; a[𝑥; 𝜉], 𝜉

]=

1

2N[𝑥; 𝜉]T ·M(𝜉)−1 · N[𝑥; 𝜉], (62)

that we call the ℱ-statistic. The ℱ-statistic depends nonlinearly on the intrinsic parameters 𝜉 ofthe signal, it does not depend on the extrinsic parameters a.

The procedure to detect the gravitational-wave signal of the form (32) and estimate its param-eters consists of two parts. The first part is to find the (local) maxima of the ℱ-statistic (62) in the

intrinsic parameters space. The ML estimators 𝜉 of the intrinsic parameters 𝜉 are those values of𝜉 for which the ℱ-statistic attains a maximum. The second part is to calculate the estimators a ofthe extrinsic parameters a from the analytic formula (61), where the matrix M and the correlations

N are calculated for the parameters 𝜉 replaced by their ML estimators 𝜉 obtained from the firstpart of the analysis. We call this procedure the maximum likelihood detection. See Section 4.6 fora discussion of the algorithms to find the (local) maxima of the ℱ-statistic.

4.1.1 Targeted searches

The ℱ-statistic can also be used in the case when the intrinsic parameters are known. An exampleof such an analysis called a targeted search is the search for a gravitational-wave signal from aknown pulsar. In this case assuming that gravitational-wave emission follows the radio timing,the phase of the signal is known from pulsar observations and the only unknown parameters ofthe signal are the amplitude (or extrinsic) parameters a [see Eq. (30)]. To detect the signal onecalculates the ℱ-statistic for the known values of the intrinsic parameters and compares it to athreshold [67]. When a statistically-significant signal is detected, one then estimates the amplitudeparameters from the analytic formulae (61).




In [109] it was shown that the maximum-likelihood ℱ-statistic can be interpreted as a Bayesfactor with a simple, but unphysical, amplitude prior (and an additional unphysical sky-positionweighting). Using a more physical prior based on an isotropic probability distribution for theunknown spin-axis orientation of emitting systems, a new detection statistic (called the ℬ-statistic)was obtained. Monte Carlo simulations for signals with random (isotropic) spin-axis orientationsshow that the ℬ-statistic is more powerful (in terms of its expected detection probability) than theℱ-statistic. A modified version of the ℱ-statistic that can be more powerful than the original onehas been studied in [20].

4.2 Signal-to-noise ratio and the Fisher matrix

The detectability of the signal ℎ(𝑡;𝜃) is determined by the signal-to-noise ratio 𝜌. In general itdepends on all the signal’s parameters 𝜃 and can be computed from [see Eq. (47)]

𝜌(𝜃) =√

(ℎ(𝑡;𝜃)|ℎ(𝑡;𝜃)). (63)

The signal-to-noise ratio for the signal (32) can be written as

𝜌(a, 𝜉) =√aT ·M(𝜉) · a, (64)

where the components of the matrix M(𝜉) are defined in Eq. (60).

The accuracy of estimation of the signal’s parameters is determined by Fisher informationmatrix Γ. The components of Γ in the case of the Gaussian noise can be computed from Eq. (58).For the signal given in Eq. (32) the signal’s parameters (collected into the vector 𝜃) split intoextrinsic and intrinsic parameters: 𝜃 = (a, 𝜉), where a = (𝑎1, . . . , 𝑎𝑛) and 𝜉 = (𝜉1, . . . , 𝜉𝑚). Itis convenient to distinguish between extrinsic and intrinsic parameter indices. Therefore, we usecalligraphic lettering to denote the intrinsic parameter indices: 𝜉𝒜, 𝒜 = 1, . . . ,𝑚. The matrix Γhas dimension (𝑛+𝑚)× (𝑛+𝑚) and it can be written in terms of four block matrices for the twosets of the parameters a and 𝜉,

Γ(a, 𝜉) =

(Γaa(𝜉) Γa𝜉(a, 𝜉)

Γa𝜉(a, 𝜉)T Γ𝜉𝜉(a, 𝜉)

), (65)

where Γaa is an 𝑛× 𝑛 matrix with components (𝜕ℎ/𝜕𝑎𝑖|𝜕ℎ/𝜕𝑎𝑗) (𝑖, 𝑗 = 1, . . . , 𝑛), Γa𝜉 is an 𝑛×𝑚matrix with components (𝜕ℎ/𝜕𝑎𝑖|𝜕ℎ/𝜕𝜉𝒜) (𝑖 = 1, . . . , 𝑛, 𝒜 = 1, . . . ,𝑚), and finally Γ𝜉𝜉 is 𝑚×𝑚matrix with components (𝜕ℎ/𝜕𝜉𝒜|𝜕ℎ/𝜕𝜉ℬ) (𝒜,ℬ = 1, . . . ,𝑚).

We introduce two families of the auxiliary 𝑛 × 𝑛 square matrices F(𝒜) and S(𝒜ℬ) (𝒜,ℬ =1, . . . ,𝑚), which depend on the intrinsic parameters 𝜉 only (the indexes 𝒜,ℬ within parenthesesmean that they serve here as the matrix labels). The components of the matrices F(𝒜) and S(𝒜ℬ)

are defined as follows:

𝐹(𝒜)𝑖𝑗(𝜉) :=

(ℎ𝑖(𝑡; 𝜉)

𝜕ℎ𝑗(𝑡; 𝜉)

𝜕𝜉𝒜

), 𝑖, 𝑗 = 1, . . . , 𝑛, 𝒜 = 1, . . . ,𝑚, (66)

𝑆(𝒜ℬ)𝑖𝑗(𝜉) :=

(𝜕ℎ𝑖(𝑡; 𝜉)

𝜕𝜉𝒜

𝜕ℎ𝑗(𝑡; 𝜉)

𝜕𝜉ℬ

), 𝑖, 𝑗 = 1, . . . , 𝑛, 𝒜,ℬ = 1, . . . ,𝑚. (67)

Making use of the definitions (60) and (66)–(67) one can write the more explicit form of the




matrices Γaa, Γa𝜉, and Γ𝜉𝜉,

Γaa(𝜉) = M(𝜉), (68)

Γa𝜉(a, 𝜉) =(F(1)(𝜉) · a · · · F(𝑚)(𝜉) · a

), (69)

Γ𝜉𝜉(a, 𝜉) =

⎛⎝ aT · S(11)(𝜉) · a · · · aT · S(1𝑚)(𝜉) · a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

aT · S(𝑚1)(𝜉) · a · · · aT · S(𝑚𝑚)(𝜉) · a

⎞⎠ . (70)

The notation introduced above means that the matrix Γa𝜉 can be thought of as a 1×𝑚 row matrixmade of 𝑛×1 column matrices F(𝒜) ·a. Thus, the general formula for the component of the matrixΓa𝜉 is (

Γa𝜉

)𝑖𝒜 =

(F(𝒜) · a

)𝑖=

𝑛∑𝑗=1

𝐹(𝒜)𝑖𝑗 𝑎𝑗 , 𝒜 = 1, . . . ,𝑚, 𝑖 = 1, . . . , 𝑛. (71)

The general component of the matrix Γ𝜉𝜉 is given by

(Γ𝜉𝜉

)𝒜ℬ = aT · S(𝒜ℬ) · a =

𝑛∑𝑖=1

𝑛∑𝑗=1

𝑆(𝒜ℬ)𝑖𝑗𝑎𝑖𝑎𝑗 , 𝒜,ℬ = 1, . . . ,𝑚. (72)

The covariance matrix C, which approximates the expected covariances of the ML estimatorsof the parameters 𝜃, is defined as Γ−1. Applying the standard formula for the inverse of a blockmatrix [90] to Eq. (65), one gets

C(a, 𝜉) =

(Caa(a, 𝜉) Ca𝜉(a, 𝜉)

Ca𝜉(a, 𝜉)T C𝜉𝜉(a, 𝜉)

), (73)

where the matrices Caa, Ca𝜉, and C𝜉𝜉 can be expressed in terms of the matrices Γaa = M, Γa𝜉, andΓ𝜉𝜉 as follows:

Caa(a, 𝜉) = M(𝜉)−1 +M(𝜉)−1 · Γa𝜉(a, 𝜉) · Γ(a, 𝜉)−1 · Γa𝜉(a, 𝜉)T ·M(𝜉)−1, (74)

Ca𝜉(a, 𝜉) = −M(𝜉)−1 · Γa𝜉(a, 𝜉) · Γ(a, 𝜉)−1, (75)

C𝜉𝜉(a, 𝜉) = Γ(a, 𝜉)−1. (76)

In Eqs. (74) – (76) we have introduced the 𝑚×𝑚 matrix:

Γ(a, 𝜉) := Γ𝜉𝜉(a, 𝜉)− Γa𝜉(a, 𝜉)T ·M(𝜉)−1 · Γa𝜉(a, 𝜉). (77)

We call the matrix Γ (which is the Schur complement of the matrix M) the projected Fisher matrix(onto the space of intrinsic parameters). Because the matrix Γ is the inverse of the intrinsic-parameter submatrix C𝜉𝜉 of the covariance matrix C, it expresses the information available aboutthe intrinsic parameters that takes into account the correlations with the extrinsic parameters.The matrix Γ is still a function of the putative extrinsic parameters.

We next define the normalized projected Fisher matrix (which is the 𝑚×𝑚 square matrix)

Γ𝑛(a, 𝜉) :=Γ(a, 𝜉)

𝜌(a, 𝜉)2, (78)

where 𝜌 is the signal-to-noise ratio. Making use of the definition (77) and Eqs. (71)–(72) we canshow that the components of this matrix can be written in the form(

Γ𝑛(a, 𝜉))𝒜ℬ =

aT · A(𝒜ℬ)(𝜉) · aaT ·M(𝜉) · a

, 𝒜,ℬ = 1, . . . ,𝑚, (79)




where A(𝒜ℬ) is the 𝑛× 𝑛 matrix defined as

A(𝒜ℬ)(𝜉) := 𝑆(𝒜ℬ)(𝜉)− F(𝒜)(𝜉)T ·M(𝜉)−1 · F(ℬ)(𝜉), 𝒜,ℬ = 1, . . . ,𝑚. (80)

From the Rayleigh principle [90] it follows that the minimum value of the component (Γ𝑛(a, 𝜉))𝒜ℬis given by the smallest eigenvalue of the matrix M−1 ·A(𝒜ℬ). Similarly, the maximum value of the

component (Γ𝑛(a, 𝜉))𝒜ℬ is given by the largest eigenvalue of that matrix.

Because the trace of a matrix is equal to the sum of its eigenvalues, the 𝑚×𝑚 square matrixΓ with components

(Γ(𝜉))𝒜ℬ :=1

𝑛Tr(M(𝜉)−1 · A(𝒜ℬ)(𝜉)

), 𝒜,ℬ = 1, . . . ,𝑚, (81)

expresses the information available about the intrinsic parameters, averaged over the possible valuesof the extrinsic parameters. Note that the factor 1/𝑛 is specific to the case of 𝑛 extrinsic parameters.

We shall call Γ the reduced Fisher matrix. This matrix is a function of the intrinsic parametersalone. We shall see that the reduced Fisher matrix plays a key role in the signal processing theorythat we present here. It is used in the calculation of the threshold for statistically significantdetection and in the formula for the number of templates needed to do a given search.

For the case of the signal

ℎ(𝑡;𝐴0, 𝜑0, 𝜉) = 𝐴0 𝑔(𝑡; 𝜉) cos (𝜑(𝑡; 𝜉)− 𝜑0) , (82)

the normalized projected Fisher matrix Γ𝑛 is independent of the extrinsic parameters 𝐴0 and 𝜑0,and it is equal to the reduced matrix Γ [102]. The components of Γ are given by

Γ𝒜ℬ = (Γ0)𝒜ℬ − (Γ0)𝜑0𝒜(Γ0)𝜑0ℬ

(Γ0)𝜑0𝜑0

, (83)

where Γ0 is the Fisher matrix for the signal 𝑔(𝑡; 𝜉) cos (𝜑(𝑡; 𝜉)− 𝜑0).

4.3 False alarm and detection probabilities

4.3.1 False alarm and detection probabilities for known intrinsic parameters

We first present the false alarm and detection probabilities when the intrinsic parameters 𝜉 ofthe signal are known. In this case the ℱ-statistic is a quadratic form of the random variablesthat are correlations of the data. As we assume that the noise in the data is Gaussian and thecorrelations are linear functions of the data, ℱ is a quadratic form of the Gaussian random variables.Consequently the ℱ-statistic has a distribution related to the 𝜒2 distribution. One can show (seeSection III B in [65]) that for the signal given by Eq. (30), 2ℱ has a 𝜒2 distribution with 𝑛 degreesof freedom when the signal is absent and noncentral 𝜒2 distribution with 𝑛 degrees of freedom andnon-centrality parameter equal to the square of the signal-to-noise ratio when the signal is present.

As a result the pdfs 𝑝0 and 𝑝1 of the ℱ-statistic, when the intrinsic parameters are known andwhen respectively the signal is absent or present in the data, are given by

𝑝0(ℱ) =ℱ𝑛/2−1

(𝑛/2− 1)!exp(−ℱ), (84)

𝑝1(𝜌,ℱ) =(2ℱ)(𝑛/2−1)/2

𝜌𝑛/2−1𝐼𝑛/2−1

(𝜌√2ℱ)exp

(−ℱ − 1

2𝜌2), (85)




where 𝑛 is the number of degrees of freedom of 𝜒2 distribution and 𝐼𝑛/2−1 is the modified Besselfunction of the first kind and order 𝑛/2− 1. The false alarm probability 𝑃F is the probability thatℱ exceeds a certain threshold ℱ0 when there is no signal. In our case we have

𝑃F(ℱ0) :=

∫ ∞

ℱ0

𝑝0(ℱ) dℱ = exp(−ℱ0)

𝑛/2−1∑𝑘=0

ℱ0𝑘

𝑘!. (86)

The probability of detection 𝑃D is the probability that ℱ exceeds the threshold ℱ0 when a signalis present and the signal-to-noise ratio is equal to 𝜌:

𝑃D(𝜌,ℱ0) :=

∫ ∞

ℱ0

𝑝1(𝜌,ℱ) dℱ . (87)

The integral in the above formula can be expressed in terms of the generalized Marcum 𝑄-function [132, 58], 𝑃D(𝜌,ℱ0) = 𝑄(𝜌,

√2ℱ0). We see that when the noise in the detector is Gaussian

and the intrinsic parameters are known, the probability of detection of the signal depends on asingle quantity: the optimal signal-to-noise ratio 𝜌.

4.3.2 False alarm probability for unknown intrinsic parameters

Next we return to the case in which the intrinsic parameters 𝜉 are not known. Then the statisticℱ [𝑥; 𝜉] given by Eq. (62) is a certain multiparameter random process called the random field(see monographs [5, 6] for a comprehensive discussion of random fields). If the vector 𝜉 hasone component the random field is simply a random process. For random fields we define theautocovariance function 𝒞 just in the same way as we define such a function for a random process:

𝒞(𝜉, 𝜉′) := E0

[ℱ [𝑥; 𝜉]ℱ [𝑥; 𝜉′]

]− E0

[ℱ [𝑥; 𝜉]

]E0

[ℱ [𝑥; 𝜉′]

], (88)

where 𝜉 and 𝜉′ are two values of the intrinsic parameter set, and E0 is the expectation value whenthe signal is absent. One can show that for the signal (30) the autocovariance function 𝒞 is givenby

𝒞(𝜉, 𝜉′) = 1

2Tr(Q(𝜉, 𝜉′) ·M(𝜉′)−1 · Q(𝜉, 𝜉′)T ·M(𝜉)−1

), (89)

where Q is an 𝑛× 𝑛 matrix with components

Q(𝜉, 𝜉′)𝑖𝑗 :=(ℎ𝑖(𝑡; 𝜉)|ℎ𝑗(𝑡; 𝜉

′)), 𝑖, 𝑗 = 1, . . . , 𝑛. (90)

Obviously Q(𝜉, 𝜉) = M(𝜉), therefore 𝒞(𝜉, 𝜉) = 𝑛/2.One can estimate the false alarm probability in the following way [68]. The autocovariance

function 𝒞 tends to zero as the displacement Δ𝜉 = 𝜉′−𝜉 increases (it is maximal for Δ𝜉 = 0). Thuswe can divide the parameter space into elementary cells such that in each cell the autocovariancefunction 𝒞 is appreciably different from zero. The realizations of the random field within a cell willbe correlated (dependent), whereas realizations of the random field within each cell and outside ofthe cell are almost uncorrelated (independent). Thus, the number of cells covering the parameterspace gives an estimate of the number of independent realizations of the random field.

We choose the elementary cell with its origin at the point 𝜉 to be a compact region withboundary defined by the requirement that the autocovariance 𝒞(𝜉, 𝜉′) between the origin 𝜉 andany point 𝜉′ at the cell’s boundary equals half of its maximum value, i.e., 𝒞(𝜉, 𝜉)/2. Thus, theelementary cell is defined by the inequality

𝒞(𝜉, 𝜉′) ≤ 1

2𝒞(𝜉, 𝜉) = 𝑛

4, (91)




with 𝜉 at the cell’s center and 𝜉′ on the cell’s boundary.To estimate the number of cells we perform the Taylor expansion of the autocovariance function

up to the second-order terms:

𝒞(𝜉, 𝜉′) ∼=𝑛

2+

𝑚∑𝒜=1

𝜕𝒞(𝜉, 𝜉′)𝜕𝜉′𝒜

𝜉′=𝜉

Δ𝜉𝒜 +1

2

𝑚∑𝒜,ℬ=1

𝜕2𝒞(𝜉, 𝜉′)𝜕𝜉′𝒜 𝜕𝜉′ℬ

𝜉′=𝜉

Δ𝜉𝒜 Δ𝜉ℬ. (92)

As 𝒞 attains its maximum value when 𝜉 − 𝜉′ = 0, we have

𝜕𝒞(𝜉, 𝜉′)𝜕𝜉′𝒜

𝜉′=𝜉

= 0, 𝒜 = 1, . . . ,𝑚. (93)

Let us introduce the symmetric matrix G with components

𝐺𝒜ℬ(𝜉) := − 1

2 𝒞(𝜉, 𝜉)𝜕2𝒞(𝜉, 𝜉′)𝜕𝜉′𝒜 𝜕𝜉′ℬ

𝜉′=𝜉

, 𝒜,ℬ = 1, . . . ,𝑚. (94)

Then, the inequality (91) for the elementary cell can approximately be written as

𝑚∑𝒜,ℬ=1

𝐺𝒜ℬ(𝜉)Δ𝜉𝒜 Δ𝜉ℬ ≤ 1

2. (95)

It is interesting to find a relation between the matrix 𝐺 and the Fisher matrix. One can show(see [78], Appendix B) that the matrix 𝐺 is precisely equal to the reduced Fisher matrix Γ givenby Eq. (81).

If the components of the matrix G are constant (i.e., they are independent of the values of theintrinsic parameters 𝜉 of the signal), the above equation defines a hyperellipsoid in 𝑚-dimensional(𝑚 is the number of the intrinsic parameters) Euclidean space R𝑚. The 𝑚-dimensional Euclideanvolume 𝑉cell of the elementary cell given by Eq. (95) equals

𝑉cell =(𝜋/2)𝑚/2

Γ(𝑚/2 + 1)√detG

, (96)

where Γ denotes the Gamma function. We estimate the number 𝑁cells of elementary cells bydividing the total Euclidean volume 𝑉 of the 𝑚-dimensional intrinsic parameter space by thevolume 𝑉cell of one elementary cell, i.e., we have

𝑁cells =𝑉

𝑉cell. (97)

The components of the matrix G are constant for the signal ℎ(𝑡;𝐴0, 𝜑0, 𝜉) = 𝐴0 cos (𝜑(𝑡; 𝜉)− 𝜑0),provided the phase 𝜑(𝑡; 𝜉) is a linear function of the intrinsic parameters 𝜉.

To estimate the number of cells in the case when the components of the matrix G are notconstant, i.e., when they depend on the values of the intrinsic parameters 𝜉, one replaces Eq. (97)by

𝑁cells =Γ(𝑚/2 + 1)

(𝜋/2)𝑚/2

∫𝑉

√detG(𝜉) d𝑉. (98)

This formula can be thought of as interpreting the matrix G as the metric on the parameter space.This interpretation appeared for the first time in the context of gravitational-wave data analysisin the work by Owen [102], where an analogous integral formula was proposed for the number oftemplates needed to perform a search for gravitational-wave signals from coalescing binaries.




The concept of number of cells was introduced in [68] and it is a generalization of the idea ofan effective number of samples introduced in [46] for the case of a coalescing binary signal.

We approximate the pdf of the ℱ-statistic in each cell by the pdf 𝑝0(ℱ) of the ℱ-statistic whenthe parameters are known [it is given by Eq. (84)]. The values of the ℱ-statistic in each cell can beconsidered as independent random variables. The probability that ℱ does not exceed the thresholdℱ0 in a given cell is 1− 𝑃F(ℱ0), where 𝑃F(ℱ0) is given by Eq. (86). Consequently the probabilitythat ℱ does not exceed the threshold ℱ0 in all the 𝑁cells cells is [1 − 𝑃F(ℱ0)]

𝑁cells . Thus, theprobability 𝑃𝑇

F that ℱ exceeds ℱ0 in one or more cells is given by

𝑃𝑇F (ℱ0) = 1− [1− 𝑃F(ℱ0)]

𝑁cells . (99)

By definition, this is the false alarm probability when the phase parameters are unknown. Thenumber of false alarms 𝑁F is given by

𝑁F = 𝑁cells𝑃𝑇F (ℱ0). (100)

A different approach to the calculation of the number of false alarms using the Euler characteristicof level crossings of a random field is described in [65].

It was shown (see [39]) that for any finite ℱ0 and 𝑁cells, Eq. (99) provides an upper bound forthe false alarm probability. Also in [39] a tighter upper bound for the false alarm probability wasderived by modifying a formula obtained by Mohanty [92]. The formula amounts essentially tointroducing a suitable coefficient multiplying the number 𝑁cells of cells.

4.3.3 Detection probability for unknown intrinsic parameters

When the signal is present in the data a precise calculation of the pdf of the ℱ-statistic is verydifficult because the presence of the signal makes the data’s random process non-stationary. As afirst approximation we can estimate the probability of detection of the signal when the intrinsic pa-rameters are unknown by the probability of detection when these parameters are known [it is givenby Eq. (87)]. This approximation assumes that when the signal is present the true values of theintrinsic parameters fall within the cell where the ℱ-statistic has a maximum. This approximationwill be the better the higher the signal-to-noise ratio 𝜌 is.

4.4 Number of templates

To search for gravitational-wave signals we evaluate the ℱ-statistic on a grid in parameter space.The grid has to be sufficiently fine such that the loss of signals is minimized. In order to estimatethe number of points of the grid, or in other words the number of templates that we need to searchfor a signal, the natural quantity to study is the expectation value of the ℱ-statistic when thesignal is present.

Thus, we assume that the data 𝑥 contains the gravitational-wave signal ℎ(𝑡;𝜃) defined inEq. (32), so 𝑥(𝑡;𝜃) = ℎ(𝑡;𝜃) + 𝑛(𝑡). The parameters 𝜃 = (a, 𝜉) of the signal consist of extrinsicparameters a and intrinsic parameters 𝜉. The data 𝑥 will be correlated with the filters ℎ𝑖(𝑡; 𝜉

′)(𝑖 = 1, . . . , 𝑛) parameterized by the values 𝜉′ of the intrinsic parameters. The ℱ-statistic can thusbe written in the form [see Eq. (62)]

ℱ [𝑥(𝑡; a, 𝜉); 𝜉′] =1

2N[𝑥(𝑡; a, 𝜉); 𝜉′]T ·M(𝜉′)−1 · N[𝑥(𝑡; a, 𝜉); 𝜉′], (101)

where the matrices M and N are defined in Eqs. (60). The expectation value of the ℱ-statistic (101)is

E[ℱ [𝑥(𝑡; a, 𝜉); 𝜉′]

]=

1

2

(𝑛+ aT · Q(𝜉, 𝜉′) ·M(𝜉′)−1 · Q(𝜉, 𝜉′)T · a

), (102)




where the matrix Q is defined in Eq. (90). Let us rewrite the expectation value (102) in thefollowing form,

E[ℱ [𝑥(𝑡; a, 𝜉); 𝜉′]

]=

1

2

(𝑛+ 𝜌(a, 𝜉)2𝒞n(a, 𝜉, 𝜉′)

), (103)

where 𝜌 is the signal-to-noise ratio and where we have introduced the normalized correlation func-tion 𝒞n,

𝒞n(a, 𝜉, 𝜉′) :=aT · Q(𝜉, 𝜉′) ·M(𝜉′)−1 · Q(𝜉, 𝜉′)T · a

aT ·M(𝜉) · a. (104)

From the Rayleigh principle [90] it follows that the minimum value of the normalized correlationfunction is equal to the smallest eigenvalue of the matrix M(𝜉)−1 · Q(𝜉, 𝜉′) · M(𝜉′)−1 · Q(𝜉, 𝜉′)T,whereas the maximum value is given by its largest eigenvalue. We define the reduced correlationfunction 𝒞 as

𝒞(𝜉, 𝜉′) := 1

2Tr(M(𝜉)−1 · Q(𝜉, 𝜉′) ·M(𝜉′)−1 · Q(𝜉, 𝜉′)T

). (105)

As the trace of a matrix equals the sum of its eigenvalues, the reduced correlation function 𝒞 isequal to the average of the eigenvalues of the normalized correlation function 𝒞n. In this sensewe can think of the reduced correlation function as an “average” of the normalized correlationfunction. The advantage of the reduced correlation function is that it depends only on the intrinsicparameters 𝜉, and thus is suitable for studying the number of grid points on which the ℱ-statisticneeds to be evaluated. We also note that the normalized correlation function 𝒞 precisely coincideswith the autocovariance function 𝒞 of the ℱ-statistic given by Eq. (89).

As in the calculation of the number of cells in order to estimate the number of templates weperform a Taylor expansion of 𝒞 up to second order terms around the true values of the parameters,and we obtain an equation analogous to Eq. (95),

𝑚∑𝒜,ℬ=1

G𝒜ℬ Δ𝜉𝒜 Δ𝜉ℬ = 1− 𝐶0, (106)

where G is given by Eq. (94). By arguments identical to those in deriving the formula for thenumber of cells we arrive at the following formula for the number of templates:

𝑁t =1

(1− 𝐶0)𝑚/2

Γ(𝑚/2 + 1)

𝜋𝑚/2

∫𝑉

√detG(𝜉) d𝑉. (107)

When 𝐶0 = 1/2 the above formula coincides with the formula for the number 𝑁cells of cells,Eq. (98). Here we would like to place the templates sufficiently closely so that the loss of signals isminimized. Thus 1− 𝐶0 needs to be chosen sufficiently small. The formula (107) for the numberof templates assumes that the templates are placed in the centers of hyperspheres and that thehyperspheres fill the parameter space without holes. In order to have a tiling of the parameterspace without holes we can place the templates in the centers of hypercubes, which are inscribedin the hyperspheres. Then the formula for the number of templates reads

𝑁t =1

(1− 𝐶0)𝑚/2

𝑚𝑚/2

2𝑚

∫𝑉

√detG(𝜉) d𝑉. (108)

For the case of the signal given by Eq. (34) our formula for the number of templates is equivalentto the original formula derived by Owen [102]. Owen [102] has also introduced a geometric approachto the problem of template placement involving the identification of the Fisher matrix with a metricon the parameter space. An early study of the template placement for the case of coalescingbinaries can be found in [121, 45, 26]. Applications of the geometric approach of Owen to the caseof spinning neutron stars and supernova bursts are given in [33, 16].




4.4.1 Covering problem

The problem of how to cover the parameter space with the smallest possible number of templates,such that no point in the parameter space lies further away from a grid point than a certaindistance, is known in mathematical literature as the covering problem [38]. This was first studiedin the context of gravitational-wave data analysis by Prix [111]. The maximum distance of anypoint to the next grid point is called the covering radius 𝑅. An important class of coverings arelattice coverings. We define a lattice in 𝑚-dimensional Euclidean space R𝑚 to be the set of pointsincluding 0 such that if 𝑢 and 𝑣 are lattice points, then also 𝑢+ 𝑣 and 𝑢− 𝑣 are lattice points. Thebasic building block of a lattice is called the fundamental region. A lattice covering is a coveringof R𝑚 by spheres of covering radius 𝑅, where the centers of the spheres form a lattice. The mostimportant quantity of a covering is its thickness Θ defined as

Θ :=volume of one 𝑚-dimensional sphere

volume of the fundamental region. (109)

In the case of a two-dimensional Euclidean space the best covering is the hexagonal covering and itsthickness ≃ 1.21. For dimensions higher than 2 the best covering is not known. However, we knowthe best lattice covering for dimensions 𝑚 ≤ 23. These are 𝐴*

𝑚 lattices, which have thicknessesΘ𝐴*

𝑚equal to

Θ𝐴*𝑚= 𝑉𝑚

√𝑚+ 1

(𝑚(𝑚+ 2)

12(𝑚+ 1)

)𝑚/2

, (110)

where 𝑉𝑚 is the volume of the 𝑚-dimensional sphere of unit radius. The advantage of an 𝐴*𝑚

lattice over the hypercubic lattice grows exponentially with the number of dimensions.

For the case of gravitational-wave signals from spinning neutron stars a 3-dimensional grid wasconstructed [18]. It consists of prisms with hexagonal bases. Its thickness is around 1.84, which ismuch better than the cubic grid with a thickness of approximately 2.72. It is worse than the best3-dimensional lattice covering, which has a thickness of around 1.46.

In [19] a grid was constructed in the 4-dimensional parameter space spanned by frequency,frequency derivative, and sky position of the source, for the case of an almost monochromaticgravitational-wave signal originating from a spinning neutron star. The starting point of theconstruction was an 𝐴*

4 lattice of thickness ≃ 1.77. The grid was then constrained so that thenodes of the grid coincide with Fourier frequencies. This allowed the use of a fast Fourier transform(FFT) to evaluate the maximum-likelihood ℱ-statistic efficiently (see Section 4.6.2). The resultinglattice is only 20% thicker than the optimal 𝐴*

4 lattice.

Efficient 2-dimensional banks of templates suitable for directed searches (in which one assumesthat the position of the gravitational-wave source in the sky is known, but one does not assumethat the wave’s frequency and its derivative are a priori known) were constructed in [104]. All gridsfound in [104] enable usage of the FFT algorithm in the computation of the ℱ-statistic; they havethicknesses 0.1 – 16% larger than the thickness of the optimal 2-dimensional hexagonal covering. Inthe construction of grids the dependence on the choice of the position of the observational intervalwith respect to the origin of time axis was employed. Also the usage of the FFT algorithms withnonstandard frequency resolutions achieved by zero padding or folding the data was discussed.

The above template placement constructions are based on a Fisher matrix with constant co-efficients, i.e., they assume that the parameter manifold is flat. The generalization to curvedRiemannian parameter manifolds is difficult. An interesting idea to overcome this problem is touse stochastic template banks where a grid in the parameter space is randomly generated by somealgorithm [89, 57, 86, 119].




4.5 Suboptimal filtering

To extract gravitational-wave signals from the detector’s noise one very often uses filters that arenot optimal. We may have to choose an approximate, suboptimal filter because we do not knowthe exact form of the signal (this is almost always the case in practice) or in order to reducethe computational cost and to simplify the analysis. In the case of the signal of the form givenin Eq. (32) the most natural and simplest way to proceed is to use as detection statistic the ℱ-statistic where the filters ℎ′

𝑘(𝑡; 𝜁) (𝑘 = 1, . . . , 𝑛) are the approximate ones instead of the optimalones ℎ𝑘(𝑡; 𝜉) (𝑘 = 1, . . . , 𝑛) matched to the signal. In general the functions ℎ′

𝑘(𝑡; 𝜁) will be differentfrom the functions ℎ𝑘(𝑡; 𝜉) used in optimal filtering, and also the set of parameters 𝜁 will bedifferent from the set of parameters 𝜉 in optimal filters. We call this procedure the suboptimalfiltering and we denote the suboptimal statistic by ℱs. It is defined as [see Eq. (62)]

ℱs[𝑥; 𝜁] :=1

2Ns[𝑥; 𝜁]

T ·Ms(𝜁)−1 · Ns[𝑥; 𝜁], (111)

where the data-dependent 𝑛×1 column matrix Ns and the square 𝑛×𝑛 matrix Ms have components[see Eq. (60)]

𝑁s 𝑖[𝑥; 𝜁] := (𝑥(𝑡)|ℎ′𝑖(𝑡; 𝜁)), 𝑀s 𝑖𝑗(𝜁) := (ℎ′

𝑖(𝑡; 𝜁)|ℎ′𝑗(𝑡; 𝜁)), 𝑖, 𝑗 = 1, . . . , 𝑛. (112)

We need a measure of how well a given suboptimal filter performs. To find such a measure wecalculate the expectation value of the suboptimal statistic ℱs in the case where the data containsthe gravitational-wave signal, i.e., when 𝑥(𝑡; a, 𝜉) = 𝑛(𝑡) + ℎ(𝑡; a, 𝜉). We get

E[ℱs[𝑥(𝑡; a, 𝜉); 𝜁]

]=

1

2

(𝑛+ aT · Qs(𝜉, 𝜁) ·Ms(𝜁)

−1 · Qs(𝜉, 𝜁)T · a

), (113)

where we have introduced the matrix Qs with components

𝑄s 𝑖𝑗(𝜉, 𝜁) :=(ℎ𝑖(𝑡; 𝜉)|ℎ′

𝑗(𝑡; 𝜁)), 𝑖, 𝑗 = 1, . . . , 𝑛. (114)

Let us rewrite the expectation value (113) in the following form,

E[ℱs[𝑥(𝑡; a, 𝜉); 𝜁]

]=

1

2

(𝑛+ 𝜌(a, 𝜉)2

aT · Qs(𝜉, 𝜁) ·Ms(𝜁)−1 · Qs(𝜉, 𝜁)

T · aaT ·M(𝜉) · a

), (115)

where 𝜌 is the optimal signal-to-noise ratio [given in Eq. (64)]. This expectation value reachesits maximum equal to (𝑛 + 𝜌2)/2 when the filter is perfectly matched to the signal. Therefore, anatural measure of the performance of a suboptimal filter is the quantity FF defined by

FF(𝜉) := max(a,𝜁)

√aT · Qs(𝜉, 𝜁) ·Ms(𝜁)−1 · Qs(𝜉, 𝜁)T · a

aT ·M(𝜉) · a. (116)

We call the quantity FF the generalized fitting factor. From the Rayleigh principle, it follows thatthe generalized fitting factor is the maximum of the largest eigenvalue of the matrix M(𝜉)−1 ·Qs(𝜉, 𝜁) ·Ms(𝜁)

−1 · Qs(𝜉, 𝜁)T over the intrinsic parameters of the signal.

In the case of a gravitational-wave signal given by

𝑠(𝑡;𝐴0, 𝜉) = 𝐴0 ℎ(𝑡; 𝜉), (117)

the generalized fitting factor defined above reduces to the fitting factor introduced by Aposto-latos [13]:

FF(𝜉) = max𝜁

(ℎ(𝑡; 𝜉)|ℎ′(𝑡; 𝜁))√(ℎ(𝑡; 𝜉)|ℎ(𝑡; 𝜉))

√(ℎ′(𝑡; 𝜁)|ℎ′(𝑡; 𝜁))

. (118)




The fitting factor is the ratio of the maximal signal-to-noise ratio that can be achieved withsuboptimal filtering to the signal-to-noise ratio obtained when we use a perfectly matched, optimalfilter. We note that for the signal given by Eq. (117), FF is independent of the value of theamplitude 𝐴0.

For the case of a signal of the form

𝑠(𝑡;𝐴0, 𝜑0, 𝜉) = 𝐴0 cos (𝜑(𝑡; 𝜉) + 𝜑0) , (119)

where 𝜑0 is a constant phase, the maximum over 𝜑0 in Eq. (118) can be obtained analytically. More-over, assuming that over the bandwidth of the signal the spectral density of the noise is constantand that over the observation time cos𝜑(𝑡; 𝜉) oscillates rapidly, the fitting factor is approximatelygiven by

FF(𝜉) ∼= max𝜁

⎡⎣(∫ 𝑇0

0

cos(𝜑(𝑡; 𝜉)− 𝜑′(𝑡; 𝜁)

)d𝑡

)2

+

(∫ 𝑇0

0

sin(𝜑(𝑡; 𝜉)− 𝜑′(𝑡; 𝜁)

)d𝑡

)2⎤⎦1/2

. (120)

In designing suboptimal filters one faces the issue of how small a fitting factor one can accept.A popular rule of thumb is accepting FF = 0.97. Assuming that the amplitude of the signal andconsequently the signal-to-noise ratio decreases inversely proportionally to the distance from thesource this corresponds to 10% loss of the signals that would be detected by a matched filter.

Proposals for good suboptimal (search) templates for the case of coalescing binaries are givenin [35, 134] and for the case-spinning neutron stars in [65, 18].

4.6 Algorithms to calculate the ℱ-statistic

4.6.1 The two-step procedure

In order to detect signals we search for threshold crossings of the ℱ-statistic over the intrinsicparameter space. Once we have a threshold crossing we need to find the precise location ofthe maximum of ℱ in order to estimate accurately the parameters of the signal. A satisfactoryprocedure is the two-step procedure. The first step is a coarse search where we evaluate ℱ ona coarse grid in parameter space and locate threshold crossings. The second step, called a finesearch, is a refinement around the region of parameter space where the maximum identified by thecoarse search is located.

There are two methods to perform the fine search. One is to refine the grid around the thresholdcrossing found by the coarse search [94, 92, 134, 127], and the other is to use an optimization routineto find the maximum of ℱ [65, 78]. As initial values to the optimization routine we input the valuesof the parameters found by the coarse search. There are many maximization algorithms available.One useful method is the Nelder–Mead algorithm [79], which does not require computation of thederivatives of the function being maximized.

4.6.2 Evaluation of the ℱ-statistic

Usually the grid in parameter space is very large and it is important to calculate the optimumstatistic as efficiently as possible. In special cases the ℱ-statistic given by Eq. (62) can be fur-ther simplified. For example, in the case of coalescing binaries ℱ can be expressed in terms ofconvolutions that depend on the difference between the time-of-arrival (TOA) of the signal andthe TOA parameter of the filter. Such convolutions can be efficiently computed using FFTs. Forcontinuous sources, like gravitational waves from rotating neutron stars observed by ground-based




detectors [65] or gravitational waves form stellar mass binaries observed by space-borne detec-tors [78], the detection statistic ℱ involves integrals of the general form∫ 𝑇0

0

𝑥(𝑡)𝑚(𝑡;𝜔, 𝜉) exp(𝑖𝜔𝜑mod(𝑡; 𝜉)

)exp(𝑖𝜔𝑡) d𝑡, (121)

where 𝜉 are the intrinsic parameters excluding the frequency parameter 𝜔, 𝑚 is the amplitudemodulation function, and 𝜔𝜑mod the phase modulation function. The amplitude modulation func-tion is slowly varying compared to the exponential terms in the integral (121). We see that theintegral (121) can be interpreted as a Fourier transform (and computed efficiently with an FFT), if𝜑mod = 0 and if 𝑚 does not depend on the frequency 𝜔. In the long-wavelength approximation theamplitude function 𝑚 does not depend on the frequency. In this case, Eq. (121) can be convertedto a Fourier transform by introducing a new time variable 𝑡b [124],

𝑡b(𝑡; 𝜉) := 𝑡+ 𝜑mod(𝑡; 𝜉). (122)

Thus, in order to compute the integral (121), for each set of the intrinsic parameters 𝜉 we multiplythe data by the amplitude modulation function 𝑚, resample according to Eq. (122), and performthe FFT. In the case of LISA detector data when the amplitude modulation𝑚 depends on frequencywe can divide the data into several band-passed data sets, choosing the bandwidth for each set to besufficiently small so that the change of 𝑚 exp(𝑖𝜔𝜑mod) is small over the band. In the integral (121)we can then use as the value of the frequency in the amplitude and phase modulation function themaximum frequency of the band of the signal (see [78] for details).

4.7 Accuracy of parameter estimation

4.7.1 Fisher-matrix-based assessments

Fisher matrix has been extensively used to assess the accuracy of estimation of astrophysically-interesting parameters of different gravitational-wave signals. For ground-based interferometricdetectors, the first calculations of the Fisher matrix concerned gravitational-wave signals frominspiralling compact binaries (made of neutron stars or black holes) in the leading-order quadrupoleapproximation [50, 76, 63] and from quasi-normal modes of Kerr black hole [48].

Cutler and Flanagan [41] initiated the study of the implications of the higher-order post-Newtonian (PN) phasing formula as applied to the parameter estimation of inspiralling binarysignals. They used the 1.5PN phasing formula to investigate the problem of parameter estimation,both for spinning and non-spinning binaries, and examined the effect of the spin-orbit coupling onthe estimation of parameters. The effect of the 2PN phasing formula was analyzed independentlyby Poisson and Will [106] and Krolak, Kokkotas and Schafer [75]. In both cases the focus was tounderstand the leading-order spin-spin coupling term appearing at the 2PN level when the spinswere aligned perpendicularly to the orbital plane. Compared to [75], [106] also included a prioriinformation about the magnitude of the spin parameters, which then leads to a reduction in therms errors in the estimation of mass parameters. The case of a 3.5PN phasing formula was studiedin detail by Arun et al. [17]. Inclusion of 3.5PN effects leads to an improved estimate of the binaryparameters. Improvements are relatively smaller for lighter binaries. More recently the Fishermatrix was employed to assess the errors in estimating the parameters of nonspinning black-holebinaries using the complete inspiral-merger-ring-down waveforms [7].

Various authors have investigated the accuracy with which the LISA detector can determinebinary parameters including spin effects. Cutler [40] determined LISA’s angular resolution andevaluated the errors of the binary masses and distance considering spins aligned or anti-alignedwith the orbital angular momentum. Hughes [60] investigated the accuracy with which the redshift




can be estimated (if the cosmological parameters are derived independently), and considered theblack-hole ring-down phase in addition to the inspiralling signal. Seto [128] included the effect offinite armlength (going beyond the long wavelength approximation) and found that the accuracy ofthe distance determination and angular resolution improve. This happens because the response ofthe instrument when the armlength is finite depends strongly on the location of the source, whichis tightly correlated with the distance and the direction of the orbital angular momentum. Vec-chio [140] provided the first estimate of parameters for precessing binaries when only one of the twosupermassive black holes carries spin. He showed that modulational effects decorrelate the binaryparameters to some extent, resulting in a better estimation of the parameters compared to the casewhen spins are aligned or antialigned with orbital angular momentum. Hughes and Menou [61]studied a class of binaries, which they called “golden binaries,” for which the inspiral and ring-downphases could be observed with good enough precision to carry out valuable tests of strong-fieldgravity. Berti, Buonanno and Will [29] have shown that inclusion of non-precessing spin-orbit andspin-spin terms in the gravitational-wave phasing generally reduces the accuracy with which theparameters of the binary can be estimated. This is not surprising, since the parameters are highlycorrelated, and adding parameters effectively dilutes the available information.

Extensive study of accuracy of parameter estimation for continuous gravitational-wave signalsfrom spinning neutron stars was performed in [64]. In [129] Seto used the Fisher matrix to studythe possibility of determining distances to rapidly rotating isolated neutron stars by measuring thecurvature of the wave fronts.

4.7.2 Comparison with the Cramer–Rao bound

In order to test the performance of the maximization method of the ℱ-statistic it is useful toperform Monte Carlo simulations of the parameter estimation and compare the simulated variancesof the estimators with the variances calculated from the Fisher matrix. Such simulations wereperformed for various gravitational-wave signals [73, 26, 65, 36]. In these simulations one observesthat, above a certain signal-to-noise ratio, called the threshold signal-to-noise ratio, the resultsof the Monte Carlo simulations agree very well with the calculations of the rms errors from theinverse of the Fisher matrix. However, below the threshold signal-to-noise ratio they differ by alarge factor. This threshold effect is well known in signal processing [139]. There exist more refinedtheoretical bounds on the rms errors that explain this effect, and they were studied in the contextof the gravitational-wave signals from coalescing binaries [98].

Use of the Fisher matrix in the assessment of accuracy of the parameter estimation has beencritically examined in [138], where a criterion has been established for the signal-to-noise ratioabove which the inverse of the Fisher matrix approximates well covariances of the parameterestimators. In [148, 142] the errors of ML estimators of parameters of gravitational-wave signalsfrom nonspinning black-hole binaries were calculated analytically using a power expansion of thebias and the covariance matrix in inverse powers of the signal-to-noise ratio. The first-order termin this covariance matrix expansion is the inverse of the Fisher information matrix. The use ofhigher-order derivatives of the likelihood function in these expansions makes the errors predictionsensitive to the secondary lobes of the pdf of the ML estimators. Conditions for the validity ofthe Cramer–Rao lower bound are discussed in [142] as well, and some new features in regions ofthe parameter space so far not explored are predicted (e.g., that the bias can become the mostimportant contributor to the parameters errors for high-mass systems with masses 200𝑀⊙ andabove).

There exists a simple model that explains the deviations from the covariance matrix and re-produces well the results of the Monte Carlo simulations (see also [25]). The model makes useof the concept of the elementary cell of the parameter space that we introduced in Section 4.3.2.The calculation given below is a generalization of the calculation of the rms error for the case of a




monochromatic signal given by Rife and Boorstyn [116].When the values of parameters of the template that correspond to the maximum of the func-

tional ℱ fall within the cell in the parameter space where the signal is present, the rms error issatisfactorily approximated by the inverse of the Fisher matrix. However, sometimes, as a resultof noise, the global maximum is in the cell where there is no signal. We then say that an outlierhas occurred. In the simplest case we can assume that the probability density of the values of theoutliers is uniform over the search interval of a parameter, and then the rms error is given by

𝜎2out =

Δ2

12, (123)

where Δ is the length of the search interval for a given parameter. The probability that an outlieroccurs will be higher the lower the signal-to-noise ratio is. Let 𝑞 be the probability that an outlieroccurs. Then the total variance 𝜎2 of the estimator of a parameter is the weighted sum of the twoerrors

𝜎2 = 𝜎2out𝑞 + 𝜎2

CR(1− 𝑞), (124)

where 𝜎CR is the rms errors calculated from the covariance matrix for a given parameter. One canshow [65] that the probability 𝑞 can be approximated by the following formula:

𝑞 = 1−∫ ∞

0

𝑝1(𝜌,ℱ)

(∫ ℱ

0

𝑝0(𝑦) d𝑦

)𝑁cells−1

dℱ , (125)

where 𝑝0 and 𝑝1 are the pdfs of the ℱ-statistic (for known intrinsic parameters) when the signal isabsent or present in data, respectively [they are given by Eqs. (84) and (85)], and where 𝑁cells isthe number of cells in the intrinsic parameter space. Eq. (125) is in good but not perfect agreementwith the rms errors obtained from the Monte Carlo simulations (see [65]). There are clearly otherreasons for deviations from the Cramer–Rao bound as well. One important effect (see [98]) isthat the functional ℱ has many local subsidiary maxima close to the global one. Thus, for a lowsignal-to-noise ratio the noise may promote the subsidiary maximum to a global one.

4.8 Upper limits

Detection of a signal is signified by a large value of the ℱ-statistic that is unlikely to arise from thenoise-only distribution. If instead the value of ℱ is consistent with pure noise with high probabilitywe can place an upper limit on the strength of the signal. One way of doing this is to take theloudest event obtained in the search and solve the equation

𝑃D(𝜌UL,ℱL) = 𝛽 (126)

for signal-to-noise ratio 𝜌UL, where 𝑃D is the detection probability given by Eq. (87), ℱL is thevalue of the ℱ-statistic corresponding to the loudest event, and 𝛽 is a chosen confidence [23, 2].Then 𝜌UL is the desired upper limit with confidence 𝛽.

When gravitational-wave data do not conform to a Gaussian probability density assumed inEq. (87), a more accurate upper limit can be obtained by injecting the signals into the detector’sdata and thereby estimating the probability of detection 𝑃D [4].




5 Network of Detectors

Several gravitational-wave detectors can observe gravitational waves from the same source. Forexample a network of bar detectors can observe a gravitational-wave burst from the same super-nova explosion, or a network of laser interferometers can detect the inspiral of the same compactbinary system. The space-borne LISA detector can be considered as a network of three detectorsthat can make three independent measurements of the same gravitational-wave signal. Simulta-neous observations are also possible among different types of detectors. For example, a search forsupernova bursts can be performed simultaneously by resonant and interferometric detectors [21].

Let us consider a network consisting of 𝑁 gravitational-wave detectors and let us denote by 𝑥𝐼

the data collected by the 𝐼th detector (𝐼 = 1, . . . , 𝑁). We assume that noises in all detectors areadditive, so the data 𝑥𝐼 is a sum of the noise 𝑛𝐼 in the 𝐼th detector and eventually a gravitational-wave signal ℎ𝐼 registered by the 𝐼th detector,

𝑥𝐼(𝑡) = 𝑛𝐼(𝑡) + ℎ𝐼(𝑡), 𝐼 = 1, . . . , 𝑁. (127)

It is convenient to collect all the data streams, all the noises, and all the gravitational-wave signalsinto column 𝑁 × 1 matrices denoted respectively by x, n, and h,

x(𝑡) :=

⎛⎜⎝𝑥1(𝑡)...

𝑥𝑁 (𝑡)

⎞⎟⎠ , n(𝑡) :=

⎛⎜⎝𝑛1(𝑡)...

𝑛𝑁 (𝑡)

⎞⎟⎠ , h(𝑡) :=

⎛⎜⎝ℎ1(𝑡)...

ℎ𝑁 (𝑡)

⎞⎟⎠ , (128)

then Eqs. (127) can shortly be written as

x(𝑡) = n(𝑡) + h(𝑡). (129)

If additionally all detectors’ noises are stationary, Gaussian, and continuous-in-time random pro-cesses with zero means, the network log likelihood function is given by

log Λ[x] = (x|h)− 1

2(h|h), (130)

where the scalar product ( · | · ) is defined by

(x|y) := 4ℜ∫ ∞

0

x(𝑓)T · S𝑛(𝑓)−1 · y(𝑓)* d𝑓. (131)

Here S𝑛 is the one-sided cross spectral density matrix of the noises of the detector network, whichis defined by (here E denotes the expectation value)

E[n(𝑓) · n*(𝑓 ′)

T]=

1

2𝛿(𝑓 − 𝑓 ′)S𝑛(|𝑓 |). (132)

The analysis is greatly simplified if the cross spectrum matrix S𝑛 is diagonal. This meansthat the noises in various detectors are uncorrelated. This is the case when the detectors of thenetwork are in widely separated locations, like, for example, the two LIGO detectors. However,this assumption is not always satisfied. An important case is the LISA detector where the noises ofthe three independent responses are correlated. Nevertheless for the case of LISA one can find a setof three combinations for which the noises are uncorrelated [108, 112]. When the cross spectrummatrix is diagonal the network log likelihood function is just the sum of the log likelihood functionsfor each detector.

Derivation of the likelihood function for an arbitrary network of detectors can be found in [49].Applications of optimal filtering for observations of gravitational-wave signals from coalescing bi-naries by networks of ground-based detectors are given in [63, 41, 62, 103], and for the case of




stellar-mass binaries observed by LISA space-borne detector in [78, 118]. The single-detector ℱ-statistic for nearly monochromatic gravitational waves from spinning neutron stars was generalizedto the case of a network of detectors (also with time-varying noise curves) in [42] (in this workthe ℱ-statistic was also generalized from the usual single-source case to the case of a collectionof known sources). The reduced Fisher matrix [defined in Eq. (81)] for the case of a network ofinterferometers observing spinning neutron stars has been derived and studied in [110].

Network searches for gravitational-wave burst signals of unknown shape are often based onmaximization of the network likelihood function over each sample of the unknown polarizationwaveforms ℎ+ and ℎ× and over sky positions of the source [52, 95]. A least-squares-fit solution forthe estimation of the sky location of the source and the polarization waveforms by a network ofthree detectors for the case of a broadband burst was obtained in [56].

There is also another important method for analyzing the data from a network of detectors – thesearch for coincidences of events among detectors. This analysis is particularly important when wesearch for supernova bursts, the waveforms of which are not very well known. Such signals can beeasily mimicked by the non-Gaussian behavior of the detector noise. The idea is to filter the dataoptimally in each of the detectors and obtain candidate events. Then one compares parametersof candidate events, like, for example, times of arrivals of the bursts, among the detectors in thenetwork. This method is widely used in the search for supernovae by networks of bar detectors [24].A new geometric coincident algorithm of combining the data from a network of detectors wasproposed in [117]. This algorithm employs the covariances between signal’s parameters in such away that it associates with each candidate event an ellipsoidal region in parameter space definedby the covariance matrix. Events from different detectors are deemed to be in coincidence if theirellipsoids have a nonzero overlap. The coincidence and the coherent strategies of multidetectordetection of gravitational-wave signals from inspiralling compact binaries have been comparedin [96, 97, 32]. [96] considered detectors in pairs located in the same site and [97] pairs ofdetectors at geographically separated sites. The case of three detectors (like the network of twoLIGO detectors and the Virgo detector) has been considered in detail in [32], where it wasdemonstrated that the hierarchical coherent pipeline on Gaussian data has a better performancethan the pipeline with just the coincident stage.

A general framework for studying the effectiveness of networks of interferometric gravitational-wave detectors has been proposed in [125]. Using this framework it was shown that adding afourth detector to the existing network of LIGO/VIRGO detectors can dramatically increase, by afactor of 2 to 4, the detected event rate by allowing coherent data analysis to reduce the spuriousinstrumental coincident background.




6 Non-stationary, Non-Gaussian, and Non-linear Data

Eqs. (61) and (62) provide maximum likelihood estimators only when the noise in which the signalis buried is Gaussian. There are general theorems in statistics indicating that the Gaussian noiseis ubiquitous. One is the central limit theorem, which states that the mean of any set of variableswith any distribution having a finite mean and variance tends to the normal distribution. Theother comes from the information theory and says that the probability distribution of a randomvariable with a given mean and variance, which has the maximum entropy (minimum information)is the Gaussian distribution. Nevertheless, analysis of the data from gravitational-wave detectorsshows that the noise in the detector may be non-Gaussian (see, e.g., Figure 6 in [22]). The noisein the detector may also be a non-linear and a non-stationary random process.

The maximum likelihood method does not require that the noise in the detector be Gaussianor stationary. However, in order to derive the optimum statistic and calculate the Fisher matrixwe need to know the statistical properties of the data. The probability distribution of the datamay be complicated, and the derivation of the optimum statistic, the calculation of the Fishermatrix components and the false alarm probabilities may be impractical. However, there is oneimportant result that we have already mentioned. The matched-filter, which is optimal for theGaussian case is also a linear filter that gives maximum signal-to-noise ratio no matter what thedistribution of the data. Monte Carlo simulations performed by Finn [49] for the case of a network ofdetectors indicate that the performance of matched-filtering (i.e., the maximum likelihood methodfor Gaussian noise) is satisfactory for the case of non-Gaussian and stationary noise.

Allen et al. [10, 11] derived an optimal (in the Neyman–Pearson sense, for weak signals) signalprocessing strategy, when the detector noise is non-Gaussian and exhibits tail terms. This strategyis robust, meaning that it is close to optimal for Gaussian noise but far less sensitive than conven-tional methods to the excess large events that form the tail of the distribution. This strategy isbased on a locally optimal test [71] that amounts to comparing a first non-zero derivative

Λ𝑛[𝑥] =d𝑛Λ[𝑥|𝜖]

d𝜖𝑛

𝜖=0

(133)

of the likelihood ratio with respect to the amplitude of the signal with a threshold instead of thelikelihood ratio itself.

The non-stationarity in the case of Gaussian and uncorrelated noise can be easily incorporatedinto matched filtering (see Appendix C of [1]). Let us assume that a noise sample 𝑛𝑙 in the datahas a Gaussian pdf with a variance 𝜎2

𝑙 and zero mean (𝑙 = 1, . . . , 𝑁 , where 𝑁 is the number of datapoints). Different noise samples may have distributions with different variances. We also assumethat the noise samples are uncorrelated, then the autocorrelation function 𝐾(𝑙, 𝑙′) of the noise isgiven by [see Eq. (39)]

𝐾(𝑙, 𝑙′) = 𝜎2𝑙 𝛿𝑙𝑙′ , (134)

where 𝛿𝑙𝑙′ is the Kronecker delta function. In the case of a known signal ℎ𝑙 and additive noise theoptimal filter 𝑞𝑙 is the solution of the following equation [which is a discrete version of the integralEq. (41)]:

ℎ𝑙 =𝑁∑

𝑙′=1

𝐾(𝑙, 𝑙′) 𝑞𝑙′ . (135)

Thus, we have the following equation for the filter 𝑞𝑙:

𝑞𝑙 =ℎ𝑙

𝜎2𝑙

, (136)




and the following expression for the log likelihood ratio:

lnΛ[𝑥] =𝑁∑𝑙=1

ℎ𝑙𝑥𝑙

𝜎2𝑙

− 1

2

𝑁∑𝑙=1

ℎ2𝑙

𝜎2𝑙

. (137)

Thus, we see that for non-stationary, uncorrelated Gaussian noise the optimal processing is identicalto matched filtering for a known signal in stationary Gaussian noise, except that we divide boththe data 𝑥𝑙 and the signal ℎ𝑙 by time-varying standard deviation of the noise. This may be thoughtof as a special case of whitening the data and then correlating it using a whitened filter.

In the remaining part of this section we review some statistical tests and methods to detectnon-Gaussianity, non-stationarity, and non-linearity in the data. A classical test for a sequenceof data to be Gaussian is the Kolmogorov–Smirnov test [37]. It calculates the maximum distancebetween the cumulative distribution of the data and that of a normal distribution, and assessesthe significance of the distance. A similar test is the Lillifors test [37], but it adjusts for the factthat the parameters of the normal distribution are estimated from the data rather than specifiedin advance. Another test is the Jarque–Bera test [69], which determines whether sample skewnessand kurtosis are unusually different from their Gaussian values.

A useful test to detect outliers in the data is Grubbs’ test [55]. This test assumes that thedata has an underlying Gaussian probability distribution but it is corrupted by some disturbances.Grubbs’ test detects outliers iteratively. Outliers are removed one by one and the test is iterateduntil no outliers are detected. Grubbs’ test is a test of the null hypothesis:

1. H0: There are no outliers in the data set 𝑥𝑘.

against the alternate hypothesis:

2. H1: There is at least one outlier in the data set 𝑥𝑘.

The Grubbs’ test statistic is the largest absolute deviation from the sample mean in units of thesample standard deviation, so it is defined as

𝐺 =max𝑘 |𝑥𝑘 − 𝜇|

𝜎, (138)

where 𝜇 and 𝜎 denote the sample mean and the sample standard deviation, respectively. Thehypothesis of no outliers is rejected if

𝐺 >𝑛− 1√

𝑛

⎯⎸⎸⎷ 𝑡2𝛼/(2𝑛),𝑛−2

𝑛− 2 + 𝑡2𝛼/(2𝑛),𝑛−2

, (139)

where 𝑡𝛼/(2𝑛),𝑛−2 denotes the critical value of the t-distribution with 𝑛− 2 degrees of freedom anda significance level of 𝛼/(2𝑛).

Grubbs’ test has been used to identify outliers in the search of Virgo data for gravitational-wavesignals from the Vela pulsar [1]. A test to discriminate spurious events due to non-stationarityand non-Gaussianity of the data from genuine gravitational-wave signals has been developed byAllen [9]. This test, called the 𝜒2 time-frequency discriminator, is applicable to the case of broad-band signals, such as those coming from compact coalescing binaries.

Let now 𝑥𝑘 and 𝑢𝑙 be two discrete-in-time random processes (−∞ < 𝑘, 𝑙 < ∞) and let 𝑢𝑙 beindependent and identically distributed (i.i.d.) random variables. We call the process 𝑥𝑘 linear ifit can be represented by

𝑥𝑘 =𝑁∑𝑙=0

𝑎𝑙𝑢𝑘−𝑙, (140)




where 𝑎𝑙 are constant coefficients. If 𝑢𝑙 is Gaussian (non-Gaussian), we say that 𝑥𝑙 is linearGaussian (non-Gaussian). In order to test for linearity and Gaussianity we examine the third-order cumulants of the data. The third-order cumulant 𝐶𝑘𝑙 of a zero mean stationary process isdefined by

𝐶𝑘𝑙 := E [𝑥𝑚𝑥𝑚+𝑘𝑥𝑚+𝑙] . (141)

The bispectrum 𝑆2(𝑓1, 𝑓2) is the two-dimensional Fourier transform of 𝐶𝑘𝑙. The bicoherence isdefined as

𝐵(𝑓1, 𝑓2) :=𝑆2(𝑓1, 𝑓2)

𝑆(𝑓1 + 𝑓2)𝑆(𝑓1)𝑆(𝑓2), (142)

where 𝑆(𝑓) is the spectral density of the process 𝑥𝑘. If the process is Gaussian, then its bispectrumand consequently its bicoherence is zero. One can easily show that if the process is linear thenits bicoherence is constant. Thus, if the bispectrum is not zero, then the process is non-Gaussian;if the bicoherence is not constant then the process is also non-linear. Consequently we have thefollowing hypothesis testing problems:

1. H1: The bispectrum of 𝑥𝑘 is nonzero.

2. H0: The bispectrum of 𝑥𝑘 is zero.

If Hypothesis 1 holds, we can test for linearity, that is, we have a second hypothesis testing problem:

3. H′1: The bicoherence of 𝑥𝑘 is not constant.

4. H′′1 : The bicoherence of 𝑥𝑘 is a constant.

If Hypothesis 4 holds, the process is linear.Using the above tests we can detect non-Gaussianity and, if the process is non-Gaussian, non-

linearity of the process. The distribution of the test statistic 𝐵(𝑓1, 𝑓2), Eq. (142), can be calculatedin terms of 𝜒2 distributions. For more details see [59].

It is not difficult to examine non-stationarity of the data. One can divide the data into shortsegments and for each segment calculate the mean, standard deviation and estimate the spectrum.One can then investigate the variation of these quantities from one segment of the data to the other.This simple analysis can be useful in identifying and eliminating bad data. Another quantity toexamine is the autocorrelation function of the data. For a stationary process the autocorrelationfunction should decay to zero. A test to detect certain non-stationarities used for analysis ofeconometric time series is the Dickey–Fuller test [34]. It models the data by an autoregressiveprocess and it tests whether values of the parameters of the process deviate from those allowed bya stationary model. A robust test for detecting non-stationarity in data from gravitational-wavedetectors has been developed by Mohanty [93]. The test involves applying Student’s t-test toFourier coefficients of segments of the data. Still another block-normal approach has been studiedby McNabb et al. [88]. It identifies places in the data stream where the characteristic statistics ofthe data change. These change points divide the data into blocks in characteristics are stationary.




7 Acknowledgments

One of us (AK) acknowledges support from the National Research Council under the Resident Re-search Associateship program at the Jet Propulsion Laboratory, California Institute of Technologyand from Max-Planck-Institut fur Gravitationsphysik. This research was supported in part by theMNiSzW grants nos. 1 P03B 029 27 and N N203 387237.




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